Do do van de dat ra la phai CO ly thuyet va bai tap ve toan roi rac de boi duong cho hpc sinh gioi toan pho thong, hpc sinh cac lop chuyen toan, chuyen tin, can boi duong can than va
Trang 1'^d gido iCu til - Thqc si Todn hoc - Gido vien chuyen Le Quy Don
Trang 2mii XURT BAN Dfll HOC QUOC Glfl Hl^ NOI
16 Hang Chuoi - Hai Ba TrUng - Ha Npi Dien t h o a i : Bien tap - Che ban: (04) 39714896;
Cty TNHH MTV DWH Khang Viet
Hanh chinh: (04) 39714899: Tona bien tap: (04) 39714897
Fax: (04) 39714899
Chiu trdch nhi^m xuat ban
Gidmdoc- Tong bien tap : TS P H A M THj T R A M
/^Dia chl 71 Dinh Tien Hoang - P Da Kao - Q 1 - TP HCM
Chiing ta nho lai hai bai toan noi tieng trong toan roi rac sau: pi
Quyet d j n h xuat bkn so: 2 9 2 L K - T N / Q D - N X B D H Q G H N , cap ng^y 09/10/2012
In xong n o p \\Ju chieu quy IV nSm 2012
- EULER va bai toan bay cau 6 KONIGSBERG
6 thanh pho KONIGSBERG (LITHUANIA) c6 7 cay cau bac ngang con song pregel nhu hinh ve tren:
Nguai ta dat ra cau do
Tim each di qua tat ca bky cay cau nay, moi cai diing mpt Ian roi quay ve
diem xuat phat
Nam 1736 Leonahard Euler (1707
- 1783) da chung minh khong the c6 mpt duong di nao nhu the bang lap
'uan sau: i^.u, •
Bieu dien bon mien dat A, B, C, D bang bon diem trong mat phang, moi
cau noi hai mien duoc bieu dien bang
nipt doan noi hai diem tuong ung ta
se CO do thi sau:
Trang 3Boi dudng hqc sink gioi Toan tohgrp - red rac, Nguyen Van Thong
Bay gio, m p t d i r a n g d i qua bay cay cau, m o i cai d u n g m p t Ian roi quay ve
diem xua't phat se c6 so' Ian d i den A bang so' Ian r o i k h o i A , nghia la phai sit
d u n g den m p t so chan cay cau noi v o i A V i so cau noi v o i A la 5 (le) nen
khong CO d u o n g d i nao thoa m a n dieu kien bai toan Y t u o n g tren cua Euler da
khai sinh nganh toan hpc c6 nhieu ap d u n g do la l i thuyet do t h i ^ , ,r
^di toan2: >5,., •f>:v,i <ori :m -^tn ,;.u:; i u-^-i
Bai toan b o h m a u H '•':•;.;]': i:n^,;piS.jjfin^f> ;:;firr, yjl^ii ^
- Tren m p t ban do bat k i , ta noi no dupe to mau, neu m o i m i e n ciia no
dupe to m p t m a u xac d i n h sao cho hai mien ke nhau c6 m a u sac khac nhau
Van de can d u n g toi thieu bao nhieu m a u de to m p t ban do bat k i
Bai toan da dupe dat ra khoang giua the k i XIX
M o h i n h toan hpc cua bai toan nay n h u sau:
Voi m o i ban do M cho trude, ta hay xay d u n g m p t do t h i G t u o n g u n g , m o i
m i e n cua M u n g v o i m p t d i n h cua G, 2 mien ciia M c6 chung m o t phan bien
neu va chi neu 2 d i n h t u o n g u n g trong G c6 canh noi De thay do t h i G nhan
dupe theo each tren la m p t do t h i phang n h u the bai toan to m a u M tro thanh
bai toan to m a u G Ngay tir k h i m o i xua't hien bai toan n g u o i ta dat gia thie't bai
toan la 4 T u y nhien, khong eo mpt chung m i n h nao d i i n g cho gia thie't nay
dupe dat ra Cho den n a m 1976 m p t n h o m cac nha khoa hpc (K.Appel,
W.Haken, J.Kock) da xay d t m g m p t l o i giai dua tren cac ke't qua do may tinh
I B M 360 cung cap (mat hang ngan gio tren may tinh) da khang d i n h gia thuyet
4 m a u la d u n g
- C l i n g v o i su phat trien, v o i toe d p nhanh ciia cong nghe thong tin, l y
thuyet to h p p va do t h i da tra thanh m p t linh v y c toan hpc quan trpng va can
thie't cho nhieu l i n h vuc khoa hpc va u n g dung
- Cac hpc sinh, ke ca thay giao van eon chua quan tarn nhieu den phan
nay: l y thuyet g i , bai tap nao la can thie't cho viee boi d u o n g HSG Toan, HSG
Tin, HSG trong t r u o n g THPT
- Chua CO cong t r i n h nao nghien cuu that sau sac va day d i i cho hpc sinh
pho thong ve cac u n g d u n g cua toan hpc r o i rac
- Mac dau ve mat lich su, toan roi rac ra d o i tu hai tram nam trudc day
trong qua trinh giai cac bai toan do, n h u n g bai toan dan gian T u lau, no van
con d u n g ngoai le cac p h u o n g h u o n g nghien euu cua cac nha Toan hpc, den
nay v i t r i cua no dupe khang d i n h trong Toan hpc, dac biet la t i n hpc hien dai
- Buoe phat trien cua toan r o i rac, da t h u dupe n h i i n g ke't qua va phat trien
m^nh vao khoang euoi t h e k i 19 va dau the'ki 20, k h i ma cac cong trinh ve to
Cty TNHH MTV DWH Khang Vi^l
p6, to h o p va l y thuyet do thi lien quan chat che v o i nhau, k h i do bp m o n Toanj rdi rac bat dau phat trien m a n h me ' " •
- Trong t r u o n g pho thong cua Vi?t N a m , hi?n nay bp m o n nay chua thuc svf dupe ehii trpng va quan tarn, chua c6 h u o n g nghien cuu de d u a toan roi rac vao giang day trong t r u o n g pho thong cho eo ke't qua N h a t la d o i v o i cac lop chuyen Toan ehuyen T i n , n h i m g lop pho thong chuyen can boi d u o n g Toan roi rac de cac em vao dai hpc, hpc bp m o n nay tot hon
- Ngay trong cac k y thi HSG quoe te^ hpc sinh Vi?t N a m ehiing ta thuong khong thanh cong v o i cac bai toan To hpp, to mau, do thj Trong tinh hinh toan hpc ngay cang phat trien, thi su ke't hpp giiia cac ITnh vuc toan hpc v o i nhau ngay cang cao va sau sac, doi hoi nguoi lam toan tuong lai phai biet noi ke't v o i cac bp m o n lai v o i nhau m o i dat ke't qua mong muon, ma bp m o n toan roi rac pho thong la buoe dau can thie't V o i t u d u y sau sac, manh me, no se la vien gach dau tien cho hpc sinh pho thong ren luyen buoe dau d i vao nghanh toan hpc va mpt trong so hp se tro thanh nha toan hpc ehuyen nghif p t u o n g lai
- Day la ke't qua t h i O l y m p i c toan quoe tecua doan hpc sinh Vi?t N a m Ian
t h i i 46 tai Mexico nam hpc 2005 (trich t u toan hpc va tuoi tre so 8/2005)
S B D ^ \ B A I l BAI 2 BAI 3 BAI 4 B A I 5 BAI 6
Trong m p t k y t h i hpc sinh gioi, cae t h i sinh phai giai 6 bai toan Biet rang hai
2 % bai toan bat k y l u o n c6 nhieu h o n - so thi sinh d u thi giai dupe ca hai bai nay
Trang 4Boi duanghQc sinh gioi Toan tohgp-rari r^c, Nguyeu //••,;
Day la ket qua ky thi Olympic toan quo'c te cua doan hpc sinh Vi^t Nam Ian
thii 47 tai Slovenia nam hpc 2006 (trich tu toan hpc va tuoi tre so'8/2006)
S B D \ B a i l Bai 2 Bai 3 Bai 4 Bai 5 Bai 6
doan Viet Nam rat thap trong ky IMO Ian thu 47 to chuc tai Slovenia 2006
Den nay ket qua IMO ciia chiing ta van chua c6 dau hieu tien bo, ma hpc
sinh van con lam tot cac bai toan Pi va Ps chuyen ve hinh hoc phang, con bai
toan 3, bai toan 6 van con rat yeu, bai toan 3 ve to hop roi rac, bai toan 6 thupc
ve so hpc Chiing ta c6 the quan sat ket qua doan Viet Nam dual day
STT Hp va ten Hp va ten PI P2 P3 P4 P5 P5 P6 P6 Tong Huy
diem chuong
1 Nguyen Ta Duy 7 7 0 6 7 0 17 Bac
2 Dau Hai Dang 7 7 3 7 7 0 31 Vang
3 Nguyen Phuong Minh 7 7 0 6 7 0 27 Bac
4 Le Quang Lam 7 7 0 5 0 0 19 Dong
5 Tran Hoang Bao Linh 7 1 0 5 7 0 20 Dong
6 Nguyen Hung Tarn 7 7 1 2 7 0 24 Bac
(Bang 3 (Trich Toan hpc va tuoi tre so 8 - 2012))
- Qua nhiing ky thi dinh cao, nhu vay, thi sinh d u thi toan la nhung hpc
sinh gioi dupe chpn Ipc ky luong nhung nhin vao ket qua thi ta nhan tha'y
- Hpc sinh Vi?t Nam rat manh ve hinh hpc phing, bat dang thuc, giai tich,
phuong trinh ham, nhung toan roi rac van la diem yeu Do do van de dat ra la
phai CO ly thuyet va bai tap ve toan roi rac de boi duong cho hpc sinh gioi toan
pho thong, hpc sinh cac lop chuyen toan, chuyen tin, can boi duong can than
va day dii de hpc sinh nam dupe ban chat van de va kha nang ling dung, theo
duoi dupe con duong toan hpc sau nay
6
Cty TNHH MTV DWH Khang Vi?t
Qua qua trinh giang day cac lop chuyen Toan + Tin va ket hpp voi chuong trinh boi duong HSG cua Bp giao due va dao tao toi viet tai lieu nay vdi nhung kinh nghif m ban than ma toi da day cong nghien euu trong suot qua trinh giang day Trong tai lieu nay, toi eiSng manh dan dua ra mot so khao sat mdi cua minh de tong quat hoa mpt so ket qua da biet
-Quyeh sach nay se dupe chia lam sau chuong r i ,
Chuong 1: Anh xa va luc lupng ciia cac tap hpp , X i ;j ; i
Chuong 2: Phep dem, to hpp, chinh hpp va cac mo rpng
Chuong 3: Trinh bay ve do thi, cac dinh nghia dinh ly can ban thuong dung cho hpc sinh chuyen Toan theo gioi han chuong trinh cua Bp giao due
Chuong 4: Trinh bay ve to mau gom nhung khai niem eo ban, cac dinh ly thuong su dung trong giai Toan pho thong theo gioi han chuong trinh ciia Bp giao due
Chuong 6; Nguyen ly Dirichlet - Djnh ly Ramsey ^ , ,, ,
- Trinh bay theo quan diem ly thuyet voi nhung chung minh dinh ly, vi du minh hpa, bai tap ap dung, bai tap tu luyen
- Bai tap chpn Ipe tu miic dp trung binh eho den kha, gioi, cac de thi toan quoe gia, quo'c te
- Chiing minh nhieu bai toan chia khoa, de nhin nhan van de don gian hon
- Cac dinh ly ve to hpp thuong su dung phuong phap anh xa, phuong phap song anh de chiing minh
- Cac bai toan ve do thi dua vao logic de suy luan va chiing minh eo gang dua ra nhieu bai toan chia khoa de hpc sinh de ap dung va van dung
Day la van de kho nhung tac gia mong muon gop phan minh vao ky nang
lam toan roi r^c cho hpc sinh chuyen toan nen khong khoi tranh sai sot Rat mong gop y ciia cac ban dong nghiep va cac em hpc sinh ,j ,, ^
Nguyen Van Thong
7
Trang 5BSi duang hgc sink gidi Toan td'ht^ - rbi rac, Nguyen Van Thdng
A N H X A , L U C L U C W G C U A C A C T A P H O T
§ 1 A N H X A •
I I O I N H N G H T A A N H X A : '
^inhnghia 1.1.1 •' ' >' '
M p t anh xa f t u X vao Y la m p t quy tac cho tirong u n g v a i m o i phan t u x e X
m p t phan tvr xac d i n h y e Y, k y hi?u f(x) » s c r - "
1.2 ANH V A T A O A N H :
^inh nghla 1.2.1
Gia s u f: X -> Y la m o t anh xa da cho, x la m o t phan t u t i i y y cua X, A la mQt
bp phan t i i y y ciia X, B la m p t bp phan tiiy y ciia Y The t h i n g u o i ta gpi:
• f(x) la anh cua x b o i f hay gia t n ciia anh xa f tai d i e m x
• f(A) = { y e Y I ton tai x e A sao cho f(x) = y ) la anh ciia A boi f
• f"^ (B) = { X e X I f(x) 6 B 1 la tao anh toan phan cua B b o i f
Dac biet v o i b e Y, f"^ ({ b }) = { x e X I f(x) = b } De d o n gian k y hieu ta viet
(b) thay cho f"^ ({ b}) va gpi la tao anh toan phan ciia b b o i f M o i phan t u
x e f"^ (b) g p i la m o t tao anh ciia b b o i f
K y h i f u £(A) la m o t dieu lam d u n g v i f(A) chi c6 nghla k h i A e X R6 rang ta
CO f ( 0 ) = 0 v o i m o i f Ta chung m i n h de dang cac quan he:
• A c f"-^ (f(A)) v o i m p i bp phan A ciia X
v » B c f ( r ^ (B)) v o i m p i bp phan B ciia Y
1.3 D O N ANH - TOAN ANH - S O N G ANH:
^inh nghla 1.3.1
A n h xa f: X -> Y la m o t d o n anh ne'u v o i m p i x, x ' e X, quan h ^ f(x) = f(x')
keo theo quan h$ x = x', hay x ^ x' keo theo f(x) f(x'), hay v o i m p i y e Y c6
nhieu nhat m p t x e X sao cho y = f(x) N g u o i ta con g p i m p t d o n anh f: X Y
la m p t anh xa m p t d o i mot
^inh nghla 1.3.2
Ta bao m p t anh xa f: X ^ Y la mpt toan anh ne'u f(X) = Y, n o i m p t each khac,
ne'u v o i m p i y e Y c6 i t nhat m p t x e X sao cho y = f(x) N g u a i ta con gpi m p t
toan anh f: X ^ Y la m p t anh xa t u X len Y
8
^inh nghla 1.3.3
Ta bao m p t anh xa f: X Y la m p t song anh hay m p t anh xa m p t do'i m p t t u
X len Y, ne'u no vira la d o n anh vira la toan anh, noi m p t each khac ne'u v o i m p i
y e Y CO m p t v a chi m p t x e X sao cho y = f (x)
Chang han anh xa d o n g nhat I x la m p t song anh v o i m p i X
D o do ta k y hieu h(gf) = (hg)f bang hgf va g p i la tich ciia ba anh xa f, g, h
Chii y rang neu f: X -> Y la m p t anh xa bat k y t h i ta c6: fix = l y f = f
'i>inhnghla 1.4.8 ' • - s:.v, ^.T
,inur-Gia s i i f: X - » Y va g: Y - » X la hai anh xa sao cho
gf = l x v a f g = l y The t h i g g p i la m p t anh xa ngupc ciia f
Tir d i n h nghia ta suy ra f cung la m p t anh xa ngupc ciia g
D i n h l y sau day cho ta biet k h i nao m p t anh xa c6 anh xa ngirpc
^inh nghla 1.4.4
A n h xa f: X Y CO m p t anh xa ngupc k h i va chi k h i f la m p t song anh
Chung mink Gia sii- f c6 m p t anh xa ngupc g: Y -> X
Dao lai, gia s u f la m p t song anh Q u y tac cho hrong u n g v o i m o i y e Y phan
t i i d u y nhat ciia f"^ (y) xac d i n h m p t anh xa g: Y X va ta thay ngay
9
Trang 6Boi duong hQc sink gidi Todn tohipp - rai rac, Nguyen Van Thdnjj
gf = I x va fg = ly
N h u vay f: X ^ Y c6 m p t anh xa ngugc k h i v a chi k h i f la song anh, va
trong t r u a n g h g p do ta c6 m p t anh xa ngugc g: Y X ciia f xac d i n h b o i >i ,
Ngoai anh xa ngugc nay, f con c6 anh xa ngugc nao khac khong? Ta c6:
<^inh ly 1.4.5 Gia su g: Y ^ X va g': Y -> X la hai anh xa ngugc ciia f: X Y
T h e t h i g = g' ^
ChicngminhTa c6: gf = I x va fg'= ly ' ' '
T u do: ^ ' g = gly = g(fg') = (gOg' = l^g' =
g'-N h u vay neu f: X ^ Y c6 anh xa ngugc t h i anh xa ngugc la d u y nha't, xac
d i n h b o i i s it; = , ' «
y ^ X, v o l X la phan ttr d u y nhalt ciia f"^ (y)
D o lam d u n g n g u a i ta cung ky hieu phan t u d u y nha't x ciia (y) bang
(y) va do do n g u a i ta k y hieu anh xa
§2 LlJC Ll/ONG CUA CAC TAP HOP
2.1 TAP HOP TUONG O JONG:
K h i ta dem Ian l u g t cac phan t u ciia mgt tap A, c6 the xay ra hai t r u a n g hgp: '
1 T a i m g t liic nao do, ta dem dugc het cac phan t i i ciia A Trong t r u o n g
h g p nay, tap A la h i i u han v a so cuoi ciing d a d e m t a i cho ta biet so l u g n g
p h a n t u c i i a A l y , • ' > v i
2 M a i m a i van con n h i i n g phan t i i ciia A chua dem toi Trong t r u a n g h g p
nay, tap A la v6 han
Ta noi hai tap h g p A v a B la t u a n g d u o n g (ve so' l u g n g ) neu giiia hai tap
h g p ay c6 the thiet lap m g t phep t u a n g l i n g 1 - 1 (tiic la c6 the anh xa 1 - 1 tap
nay len tap kia)
Cty TNHH MTV DWH Khang Vi^t
n < ^ 2 n (n = l , 2 , 3, ) Dieu dang chii y 6 day la B c A: 't
vay mgt tap v 6 han c6 the t u a n g tu
d u a n g v a i m g t b g phan thuc s u • ciia no (dieu k h o n g the xay ra doi
v a i tap h i i u han)
3 Tap cac d i e m trong khoang (0,1) t u a n g d u o n g v a i tap cac diem trong mgt khoang (a, b) bat ky: s u tuang l i n g c6 the thuc hien bang phep v i t u ( H i n h 1)
K h i hai tap h g p t u o n g d u o n g nhau, ta bao rang chiing ciing m g t luc l u g n g hay cung m g t ban so' N o i each khac luc l u g n g (hay ban so) ciia m g t tap bieu thi m o t t i n h chat chung cho no va tat ca cac tap t u o n g d u a n g v a i no D o i v a i tap hijoi han, t h i luc l u g n g (ban so) chinh la so' phan t i i cua no Thanh t h i i , luc
l u g n g (ban so) la khai niem true tiep m o rgng khai niem so' t u nhien Luc
l u g n g ciia m g t tap A t h u a n g dugc k i hieu I A I (hay cardA)
Neu m g t tap A t u o n g d u o n g voi mgt bg phan ciia m g t tap B, n h u n g khong tuong d u a n g v a i B, t h i ta noi luc l u g n g ciia A nho h a n luc l u g n g ciia B, hay lye l u g n g ciia B Ion h a n luc l u g n g ciia A va vie't I A I < IBI hay IBI > I A I
M g t van de t u nhien nay ra: neu mgt tap A t u o n g d u o n g v a i m g t bo phan ciia tap B va ngugc lai tap B ciing t u o n g d u a n g v a i m o t bg phan ciia A , thi eo the noi gi ve hai tap ay? D i n h ly sau day giai dap van de nay
2.1.1 ^inh ly (Cantor- Berntein) Neu tap A t u a n g d u o n g v a i m g t bg phan
ciia tap B v a ngugc lai tap B ciing t u o n g d u o n g v a i m g t bg phan ciia tap A t h i hai tap t u a n g d u a n g nhau
N h u vay, neu tap A t u o n g d u o n g v o i mgt
bg phan ciia tap B t h i chi c6 the I A I < IBI hoac
l A I = IBI (ta vie't l A I < IBI) N o i each khac
l A I < I B I , IBI < l A I l A I = IBI
Chung mink theo gia thiet c6 mgt phep t u o n g
li'ng 1 - 1 f giiia cac phan t u ciia A v a i cac phan
t u ciia m g t bg p h a n ciia B, va ngugc lai cung c6 H i n h 2
11
Trang 7BSi duang htfc sink gioi Todn tohgrp - rcri r^c, Nguyen Vin Thdng
mpt p h e p tirong ung 1- 1 g giiia cac p h a n tu
cua B vai cac phan tu ciia mpt bp phan ctia A s'< < ^ i i ; ' ; ;
Ta qui uoc gpi mpt phan tu x la "ho" cua phan tu y hay y la "con" cua x,
neu x e A , y G B v a y = f(x) (y l i n g voi x trong phep tuong ung f) hoac neu xe B,
y e A va y = g(x) (y ung vdi x trong phep tuong ung g)
Mpt phan tu x (thupc A hay B) gpi la "to tien" ciia mpt phan tu y, neu c6 mpt
day XI, X2, XK sao cho x la bo cua xi, xi la bo cua X2, X2 la bo ciia X3, XK la bo
ciia y Chang han trong hinh 2, x la to tien cua y dong thai x ciing la to tien ciia
XI, X2, X3; XI la to tien ciia X2, xs; y, v.v Bay gio ta hay chia A ra lam ba tap con:
Ac gom tat ca cac phan tu ciia A c6 mpt sochSn to tien, A i gom tat ca cac phan
tu ciia A CO mpt so le to tien va A~ gom tat ca cac phan tu ciia A c6 v6 so' to tien
Dong thoi ta cung theo each do chia B ra lam ba tap: Be, Bi, B~ Sau do ta xac
dinh mpt phep tuong ung cp nhu sau giua cac phan tu ciia A voi eae phan tu ciia
B: voi moi phan tu thupc Ac hay A - thi cho ung con eiia no, vai moi phan tu
De thay rang (p la mpt phep tuong ung 1 - 1 giua A va B That vay, trong
phep tuong ung do, voi moi x e A dl nhien ung mpt phan tu duy nhat y e B
Ngupc lai lay mpt phan tu x nao do thupc A i ; neu y e Bi thi bo ciia no thupc
Ac, nghia la y la eon ciia mpt phan Ai; neu y e Bi thi bo ciia no thupc Ac, nghia
la y la con ciia mpt phan tu x nao do thupc Ac; con neu y e B - thi bo' ciia no
thupc Aoo, nghia la y la con ciia mpt phan tu x nao do thupc A - Thanh thu moi
phan tu y e B ung voi mpt phan tu duy nhat x e A trong phep tuong ling (p
Vay (p la mpt phep tuong l i n g 1-1 giua A va B
Tir dinh ly nay ta suy ra rang cho truoc hai tap A, B, chi c6 the xay ra mpt
trong 4 truong hop:
1 l A I = IBI • K''^' ->niVx F]
2 l A I < IBI ^ J"r?>v
•j; 3 IBI < l A I ^iyjy^ (;• XA& •
4 A khong tuong duong voi B, hoac bo phan nao ciia B, va B cung khong
tuong duong voi bp phan nao ciia A
2.2 TAP OfM OLTOC
Trong tat ea cac tap v6 han thi tap "be nhat" (c6 lue lupng kem nhat) la tap
cac so tu nhien: N* = { 1, 2, 3, n, } Lue lupng eiia tap nay gpi la lue lupng
dem dupe, va mpi tap tuong duong voi no gpi la tap dem dupe Duong nhien,
cung CO the noi: Mpt tap dem dupe la mpt tap ma ta eo the danh so dupe cac
phan tu ciia no thanh mpt day v6 han: ai,a2,a3,an,
12 '
C t y TNHH M T V DWH Khang Vift
Vi du;Theo v i d\ 1 (a myc truoc), t a p |2, 4, 6, 2n, } la dem dupe Ta cung thay ngay rang cac t a p { 3, 6, 9 , 3 n , ), { 1 , 4, 9 , n ^ , } , v.v deu dem dupe; t a p hpp cac so nguyen (bao gom ca so nguyen duong, am, va so khong)
C l i n g la dem dupe v i c6 the viet dual dang: 4;,; ,
0,1, -1, 2, -2, 3, - 3 , n , -n,
Viec luc lupng dem dupe la lue lupng be nha't eiia cac t a p v6 han dupe xac
dinh boi hai dinh l i sau day:
^inh ly 22.1 ^ai cu t a p v6 han nao cung c6 mpt bp phan la t a p dem dupe
Chung minh; That vay, cho M la mpt tap v6 h^n Ta hay lay mpt phan tu bat
ky, ai 6 M , roi mpt phan tu a2 e M \, roi mpt phan tu as e M \, 32), v.v
Vi M v6 h?n, nen dieu do c6 the tie'p tyc mai va ta thu dupe tap dem dupe (ai, a2, as, } c M
^inh ly 2.2.2. Mpt bp phan eiia mpt tap dem dupe thi phai la him han hay
dem dupe
Chung mink That vay, cho B la mpt bp phan ciia tap dem dupe A = { ai, a2,
as, } Gpi la phan tu dau tien ciia B ma ta g a p trong day {81,82,83, }, a^^
la phan tu thu hai ciia B trong day so'do, v.v Neu trong eae so' 11,12, ••• c6 mpt
so Ion nhat, v i du ip, thi B = { , a^^a,^ ] la hihi han Con neu trai lai day
, a i 2 k e o dai v6 tan thi B = ( ,a^^ } la dem dupe ' '
^inh ly 2.2.3. Hpp ciia mpt hp huu han hay dem dupe t a p dem dupe cung
la tap dem dupe ^* "
Chung mink Cho mpt day tap dehi dupe: A i , A2, A3 Ta hay viet eae phan tu
eiia moi tap ay thanh mpt day va xep dat cac day thanh mpt bang nhu duoi day R6 rang tren moi duong cheo (c6 miii ten) chi eo mpt so hiiu han phan tu: tren duong cheo thu nhat chi c6 a n , tren duong eheo thu hai chi c6: 821,812, v.v noi
chung tren duong cheo thii n chi c6 n phan tu (eae apq ma p + q = n +1)
A i ^ ^11 ^ ^12 > ^ ^ 1 3 ^ ^ ^14
Veiy ta CO the Ian lupt danh so cac phan tu" tren duong eheo thii nhat, roi
den duong cheo thu hai, v.v
^inh ly 2.2.4. Khi them mpt tap hpp hixu han hay dem dupe vao mot tap v6
h c i n M , ta khong lam thay doi luc lupng ciia t^p M
13
Trang 8Boi duang hoc sink gidi Todn tohgrp - rcri r^c, Nguyen Van Thong
Chung minh:Cho N la tap thu dugc khi them vao M mQt tap A hiru han hay
dem dxxoc Theo Dinh ly 2.2.1 c6 the lay mot tap dem dugc B cz M Dat M ' =
M\B, ta CO M = M ' u B, N = M ' u B u A Theo dinh ly truac B u A ciang la dem
dugc, cho nen c6 the lap mot phep tuong ung 1-1 giiia B va B u A Sau do chi
can lap mot phep tuong ung 1- 1 giiia M ' va chinh M ' , ta se c6 mot phep tuong
ling 1-1 giiia M va N N h u vay M va N cung luc lugng
Theo dinh ly nay, ta thay rang tap cac diem thugc mot khoang (a, b)
tuong duong voi tap cac diem thugc doan [a, b] Vay tap cac diem thugc doan
[a, b] tuong duong voi tap cac diem tren toan duong thang
^inh ly 2.2.5 Tap tat ca cac day hiiu han c6 the thanh lap dugc voi cac phan
tu ciia mot tap dem dugc la dem dugc
Noi ro hon, cho A = { 3 1 , 8 2 , 3 3 , } la mot tap dem dugc, S la tap tat ca cac
day CO dang (a^^, HJ^ , , 3 ; ^ ) trong do m la mgt so' tu nhien bat ky, a^^ (k = 1, 2,
m) la nhiing phan t u (khong nhat thiet phan biet) ciia A Ta khang dinh
rang S la dem dugc
Chung mink Ggi Sm la tap cac day gom dung m phan t u ciia A
' V i S = U - i S „
Nen theo Dinh ly 2.2.3 chi can chung minh rang moi Sm la de'm dugc Voi
dieu do hien nhien v i rang = A Ta hay gia thiet dieu do diing voi
(hie de'm dugc), va chung minh cho S^^^
Cho ak la mgt phan t u xac dinh ciia A, va s|^+i la tap tat ca cac day so c6
dang (Hj^ ,3(2 , , 3 ; ^ , 3 k) Giiia s[^+i va ro rang c6 su tuong ung 1-1
y: (aii'ai2'aim'3k)<^(aii.ai2'-.ai^)
Ma Sn, da de'm dugc thi SJ^+i cung de'm dugc Do do S^+i ciing de'm dugc,
Ji$ qua l:Tap tat ca cac da thuc P(x) = ag + 3 j X + + 3^x" (n bat ky) vol cac
h^ so ao,3i, ,a„ huu ti, la de'm dugc
That v^y, cac da thuc do hrong ung 1-1 voi cac day so hihi ti (ao,ai, ,an)
Vi tap cac so'him ti la de'm dugc nen theo dinh ly tren, tap cac da thuc do ciing
de'm dugc
Jf| qua 2:Tap cac so dai so la de'm dugc
Mgt so dai so la nghi^m so'ciia mgt da thuc voi r\hung he so'hiiu ti Vi tap
cac da thuc nay la de'm dugc, ma moi da thuc chi c6 mgt so' hiiu han nghi^m
s6^ nen tap cac so d^i so' chi c6 the la hiru han hay de'm dugc Nhung no khong
the him h^n vi no bao ham tap cac so'tu nhien, vay no phai de'm dugc
nao khong de'm dugc khong?
^inh ly 2.3.1 Tap cac so thuc la khong de'm dugc
Chung mink Vi tap cac diem thugc doan [0, 1] tuong duong voi tap cac diem tren toan duong thSng, ta chi can chung minh rang tap cac diem thugc do^n [0,1] la khong dem dugc
Gia sir trai l^i rang tap do de'm dugc, nghla la c6 the danh so' thanh day:
X i , x 2 , X 3 , ta hay chia doan [0, 1] thanh ba doan bang nhau Trong ba doan do
phai CO mgt doan khong chua x^: cho doan ay la Ta lai chia ra ba doan bang nhau Trong ba doan do phai c6 mgt doan khong chua X 2 : cho doan ay la
A2 Ta lai chia A2 ra ba doan bang nhau, v.v Tie'p tuc mai, ta se c6 mgt day doan
A j 3 A 2 3> A 3 3
Voi do dai I A „ 1= va voi x^ g A ^ Vi I A „ I - > 0 (n -> °°) nen do la mgt
3 day doan that lai va theo nguyen ly Cantor, phai c6 mgt diem ^ chung cho tat
ca cac dogn ay Co' nhien ^ e [0, 1] Vay ^ phai triing voi mgt x^^ nao do
Nhung ^ 6 A „ voi mgi n, cho nen x^^ e A^^ Dieu nay trai voi each xay dung
cac doan A „ Do do gia thiet rSng tap cac diem thugc [0,1] de'm dugc la v6 ly Dinh ly tren cho thay rang luc lugng ciia tap so thuc Ion hon luc lugng de'm dugc Nguoi ta ggi no la luc lugng continum hay luc lugng c
©mh ly 2.3.2 (Cantor) Cho bat cu tap A nao thi tap tat ca cac bg phan cua
A ciing CO luc lugng Ion hon luc lugng cua A
Chung mink Ggi m la tap tat ca cac bg phan cua A Chang han ne'u A= {a, b, c} thi M gom c6: 0, {a}, {b}, {c}, (a, b}, {b, c}, {a, c}, A
Truoc he't ta hay chung minh rang A khong tuong duang voi M Muo'n the, gia sir trai lai rang A tuong duong voi M , va cho f la phep tuong ung 1-1 giiia hai tap ay Voi moi phan tir x e A ung mgt phan tir xac dinh cua M ma ta se ky hieu la f(x); ngugc lai, moi phan tir aia M la f(x) cho mgt phan tir xac dinh x e
A Vi f(x) la mgt phan tir cua M , tuc la mgt bg phan cua A nen c6 the xay ra
tinh trang x e f(x) Ta se ggi mgt phan tir x sao cho x e f(x) la "xa'u so" Con ne'u x i f(x) thi x ggi la "tot so" Ggi S la tap tat ca cac phan tir tot so cua A Vi
rang S e M nen S = £(xo) voi mgt XQ e A Ta thir xem XQ la xa'u so hay tot so? Ne'u XQ la xa'u so thi XQ e f( XQ ) = S, v6 ly v i S chi chira nhimg phan tir tot so
15
Trang 9Boi dumtg hpc sink gidi Todn td'hgp - red n^c, Nguyen Van Thong
Con neu X Q la tót só thi XQ e f( X Q ) = S : cung v6 ly v i X g da tót só thl phai thupc
S M a u thuan do chung to rang A khong the t u a n g d u o n g v d i M
i f Bay gio ta chimg m i n h rang A tuong d u o n g v o i mgt phan ctia M That vay,
gpi M ' la tap cac bp phan ciia A chi gom mpt phan t u d u y nhát thi M ' c M va ro
rang M ' va A tuong d u o n g nhau, v i m o i x e A c6 the cho i m g v d i tap {x} e M '
Tom lai, I M I > I A I , va d i n h l y da duac chung m i n h
N h u vay, cho truoc m o t tap bat ky, bao gio ciing c6 the thanh lap m p t tap
CO lyc l u p n g Ion hon D o do, khong c6 luc l u p n g (ban so) nao la Ion nhát cạ
2.4 Lyc LLTONG 2 " : '^v'' ' - ••AM.A H n mt,:
' De ket thiic van de luc l u p n g cac tap, ta hay t i m hieu them ve t h i i bac tren
tap cac ban sọ ; «
Cho A la m p t tap bat ky, S la tap tat ca cac tap con (bp phan) cua A
Neu A la h u u han va g o m n phan t u : ,a2'-.^n v o i m o i bp phan M ciia
A ta CO the cho u n g day ( ^ i , ^ 2 ' - ' ^ n ) t^ong do = 0 neu 3; ^ M va = 1 neu
Bj e M Phep t u o n g u n g do ro rang la 1 - 1 cho nen tap S t u o n g d u o n g v o i tap
tat ca cac day {^ị^Z'-'^n) '^o the thanh lap dupe bang n h i i n g soO va 1 N h u n g
de thay rang so day nay bang 2 " , v i m o i day g o m n só c6 the lay hai gia t r i
Vgy S g o m 2" phan t u : ISI = 2"
- M a rpng k y hieu do, ta q u i uoc m o t each tong quat rang: neu luc l u p n g ciia
A la a t h i lyc l u p n g ciia S (tap cac bp phan ciia A ) se dupe k y hỉu 2°'
§ 3 C N G D U N G A N H X A D E G I A I
M O T S O B A I T O A N R O I R A G ^ ^ '
K h o khan Ian nhát k h i giai m p t bai toan to hap la xac d i n h h u a n g d i " N e u
CO m p t song anh d i t u tap h u u han X den m p t tap h u u han Y t h i I X I = I Y I T u
y t u o n g do ta t i m dupe l a i giai cac bai toan sau:
^di todn 3.1 /
Mgt cuoc hgp c6 n nguai: Mgt so nguai trong ho khong quen hiet nhau, dong
thai cH moi cap 2 nguai khong quen nhau lai c6 dung 2 nguai quen chung, con
hai nguai quen nhau lai khong c6 nguai quen chung Chttng minh rang moi mgt
con ngupi trong cugc hgp c6 mgt so nhu nhau nhung nguai quen
Lin giai
Ta nhan thay rang n h u n g n g u d i quen A t h i khong quen nhau (neu khong
n h u vay t h i hp c6 A la n g u a i quen chung) Gia su B, , A j , A ^ (n > k) la
tap h p p tat ca n h i r n g n g u a i quen v o i A K h i do m o i m p t t r o n g cac A, k h o n g
quen v a i B, do do A; va B c6 d i i n g 2 n g u d i quen chung, m p t trong hai n g u o i
do la A ' n g u d i kia la m p t Bj nao do la m p t trong n h u n g n g u d i quen B
Mhvr vay m o i Aj dat dupe t u o n g u n g v d i mot B, nao dọ N e u i # j nghia la A j ,
A khae nhau t h i Bj, Bj la n g u d i quen ciia hp cung khae nhaụ (Khong n h u vay thi A va Bj, ed ba n g u d i quen chung la B, A j , A j ) D o t i n h d d i xiing, ro rang neu Bi va Bj la n h i i n g n g u d i khae nhau, v|y cd m p t song anh giiia n h i i n g ngudi quen A va B do d d so n g u d i quen eiia A va B n h u nhaụ
Neu D la m p t trong n h i i n g n g u d i ed mat trong eupe hpp t h i hoac anh ta quen v d i A hoac D k h o n g quen A t h i se cd 2 n g u d i quen chung va k h i do (theo chiing m i n h tren) A va D cd n g u d i quen bang so n g u d i quen ciia A va cQng bang so n g u d i quen ciia D Vay A va D ed so n g u d i quen n h u nhaụ N h u vay, mpt n g u d i bat k y trong n h i i n g n g u d i cd mat d eupc hpp cd so n g u d i quen bang
so n g u d i quen ciia A
mtodn3.2.(TrungQuoc-1997) *
Trong cac xau nhi phan c6 do dai n, goi â, /« so cac xau khong chiea 3 so lien tiep 0,1, 0 va \ so"cac xau khong chtta 4 so lien tiep 0, 0,1,1 hoac 1,1, 0,0 Chihtgminh rang: = 2ậ
Loi giai >,.-n 4 i-t •(••••Vsv™:,*.!
Ta gpi m p t xau thupe loai A neu no k h o n g chua 3 só lien tiep 0, 1 , 0 va gpi mpt xau thupe loai B neu no khong ehiia 4 so hang lien tiep 0, 0, 1, 1 hoac 1 , 1 ,
0, 0 v d i m o i xau X = ( x i , X 2 , , x „ ] , ta xay d u n g f(X) = ( y i y a - y n + i ) n h u sau:
y i = 0 , y ^ = X i + X 2 + + X i ^ _ i (mod 2) Rd rang X ehiia 3 so lien tiép 0, 1 , 0 k h i va
chi k h i f(X) chua 4 so hang lien tiep 0, 0 , 1 , 1 hoac 1 , 1 , 0, 0 tiic la X thupe loai A khi va chi k h i f(X) thupe B ' *
Vay f la m p t song anh d i tir tap cac xau loai A dp dai n den tap eac xau loai
B dp dai n + 1 ma bat dau bang 0 N h u n g t u m o i xau X thupe B ta nhan dupe
mpt xau X Cling thupe B bang each d d i cac p h a n t i i ciia X theo q u y tac 1 -> 0, 0
~> 1 nen so cac xau loai B d p dai n + 1 gap doi sócae xau loai B d p dai n + 1 ma bat dau bang sóỌ Tir do ta ed dpem
N h g n xet: t u vi|e so sanh luc l u p n g cae tap hpp, p h u a n g phap song anh cd
the giiip chiing ta d e m só phan t i i eiia mpt tap thong qua sy so sanh luc l u p n g
tap d d v d i m p t t^p khae ma ta da biét só p h a n t u eiia nọ ,»
^ d i todn 3.3 (Vd dich Ucraina - 1996) Ggi M la so cac songuyen duong viet trong h? thap phan c6 2n chit so,
^ong do CO n chit sol vd n chit so 2 ggi N la so tat cd cac soviet trong h? thap phan CO n chit so, trong do chi c6 cac chU sol, 2,3, 4 vd sochUsol hang so chU
so 2 Chiing minh rhngM = N. \'^{í^^p]\m B i N H T H U A N
17
Trang 10Lai giai
Vai moi só c6 n chiJ' só gom cac chu so 1, 2, 3, 4 va só chu so 1 bang só chu
só2, ta "nhan doi" thanh sóco 2n chu sótheo quy tac sau: dau tien, hai phien
ban cua só nay duQc viét ke nhau thanh só c6 hai chu só, sau do cac chu so 3 6
n chu sódau va cac chir s6'4 6 n chii sósau dugc doi thanh chu sol, cac chu so
3 6 n chir sósau va cac chir so 4 6 n chu sódau dugc doi thanh chu so 2 vi du:
1234142 12341421234142 ^ 12121221221112
Nhu thê ta thu dugic mot só c6 diing n chii só 1 va n chu so 2 R6 rang day
la mpt don anh De chxmg minh day la mpt song anh, ta xay dung anh xa
ngugc nhu sau: voi moi só c6 n chii só I v a n chu só 2, ta cat n chiJ só dau va n
chii sócuoi roi cpng chung theo cot voi quy tac: 1+1 = 1, 2 + 2 = 2,1+2 = 3, 2 + 1
= 4, va ta thu dugc mot só c6 n chii só gom cac chii só 1, 2, 3, 4 vai só chii só 1
Cho cacsónguyen duangn, k vain >k
Xetphep todn f dot vai bo sap thtctuX= (xi, ,x„) nhu sau: moi Idn chon k
so lien tiep tuy y trong X vd doi dau cua chung
Tim so cac ho thii tuX = (x^, ,Xj^) thoa man cac dieu kien:
(i) Moi phdn ttt cua X deu thuoc tap {0,1}
(ii) Co thethuc hien hitu han Idn phep todn fde ttt X nhan duac hg (1,1,1)
Lai giai Xet bg thii tu X = (xi, ,Xn) tuy ỵ Ta c6 2 nhan xet sau:
1 Co diing n - k + 1 nhom k sólien tiep
2 Sau mot só chan Ian thuc hien phep toan f cho mot nhom k só lien tiep
trong X thi gia tri k só do khong doị
Nhu vay, moi phuong an thuc hien hiiu han Ian phep toan f cho X tuang
ung voi mot bg nhi phan A = (ai,a2, ,an), trong do ta tinh theo modun 2 cua
so Ian thuc hỉn f cho nhom k sólien tiep (Xj.Xj+j, ,Xi+i^_i), va X tro thanh
(i-iP^i (-lfi"""^'<Xi,(-l)^2+ +ak.i^^^^ (-l)a„_k.ix„)
Tu do, de tháy moi bg A xac dinh duy nhát mgt bg X thoa dieu ki^n de bai
nen dap sóbai toan chinh la so cac bg A, tuc la 2"'"^^^
Cty TNHHMIV f H VHKhangVift
^ai todn 3.5
Chiing minh rang vai m, n, k e , m > k thi:
' - m + n + l - ^ m * - n + ^ m - l ' - n + l + - + ' - m - k % + k ,
Loi giai
Ta dém so cac bg so nguyen T = (ậ^z.-.â+n+i-k) vai
l<ai 0 2 <-<am+n+i-k + n +1 bang hai each Cach thu nhát: ta xem do la so
each chgn m + n + l - k s 6 ' t u m + n + l so nen se c6 C|^^n^.i each chgn Cach thii
hai: ta tháy so cac bg T c6 â^i_k =m + l - i la c|^":\C;,+i do do C^l^ each chgn
cac bg Ti=(ai,.-.,am-k) thoa l<ai< <â_k < m - i va Cn+j each chgn cac bg T2 =(am+2-k am.n+i-k3 thoa m + 2-i<â+2-k <-<am+n+i-k^m + n + l , tu do cho i chay tir 0 tai k (do m + l-k<an,+i_k<m + l ) ta dugc tong sóeach chgn la
ci^C^ci^T-iCi+ "+CkCl^+k • Ket qua do cho ta dpcm
^di todn 3.6 (Vd dick Trung Quoc -1994) i h
Chihtgminhrhng: f2^C^,6j;-_;^^/^^=C^2n.iyneZ'
k=0 ' > ' ' , ! '
Loi giai
Ta chgn ra n so tu 2n + 1 so nhu saụ Truoc het, tii 2n + 1 só, ta chia ra n cap
va só X Sau do buac 1: ta chgn ra k cap roi tu moi cap chgn ra mgt só, buac 2:
chgn [(n-k)/2] cap trong n- k cap con lai, ngoai ra, so x se dugc chgn néu n- k
le va khong dugc chgn néu n- k chan R6 rang buoc 1 c6 2^C^ each chgn va
buac 2 CO c||"j^'^^^^^ each chgn va ta chgn dugc tong cgng n s6^ trong do k chay
tu 0 toi n Lap luan do cho ta dpcm
Trong cac tap eon cua tap S = {1, 2, m + n +1}, de tháy c6 tap d^ng {ai,a2, ,a„+i}, (l<i<m + l ) trong do aj <a2< <an+i va an+a=n + k + l
V o i 0<k<m (do co CjJ+k each chgn n phan tu tu tap {1, 2 n + k}, ngoai ra c6
* e "bó sung" them mgt trong so 2'""'' tap eon cua tap (n + k + 1 , n + m
+11- ^ 19
Trang 11Bdi duong hpc sink gioi Todn to hap - rdi r^c, Nguyen Van Thong
Cho tap S gom tat cd cdc so"nguyen trong doan [1; 2002] Got T Id tap hap
tat cd cdc tap con khong rong cua S Vdi moi X thugc T, ki hi^u m(X) Id trung
binh cgng cdc phdn tit cua X Tinh m = ^
(Tong l a y theo tat ca cac tap X thuQC T) '^'^ • 8 6 • i t e J 5fa>^
Xay d u n g s o n g a n h f: T ^ T n h u s a u : f(X) = {2003 - x |x6X}, V X E T R6 rang
(X) + m ( f ( X ) ) = 2 0 0 3 D o d 6 : '
2;Xni[X)=2(m(X) + ( f C X ) ) ) H T | 2 0 0 3 ^ m = 2 ^ = ^
^di todn 3.9:
Hay tinh trung binh cgng tat cd cdc so Ngom 2002 chit sothoa man N: 99
vd cdc cha so"cua N thugc {1, 2,3,4, 5, 6, 7,8}
L o i g i a i I ^
G Q I M la tap cac so N thoa dieu kien de bai ,
Xay d y n g anh xa f n h u sau:
Chicng minh td"t cd cdc phdn tic cua M CO cung mdu ,
L a i giai
V o i mSi so nguyen a, dat Q so d u k h i chia a cho n va k y hi^u Zn = {0, 1, 2,
•••/ n -1) thi CO the coi phep to m a u da cho xac dinh anh xa
f : Z n > Z2 = {trSng, xanh)
Dieu k i l n 1) cho i) f(a) = f ( - a ) , Va e Z , ^ Dieu kien 2) va i) cho ii) f(a) = f ( a ^ ) , Va e Z
21
Trang 12Boi duang hgc sink gioi Todn to hap - rcri rac, Nguyen Van Thong
Vay vai mpi so nguyen m do ii) c6: f((m + l)k) = f[(m + k)k - k) = f(mk)
Vi (k, n) = 1 t?ip {mk/m € Z} = Zn
Vay f(Zn) la tap chi c6 1 phan tu
^ d i todn 4.2: Trong mgt hgi nghf todn hgc quoc tetochdc tai quoc gia X, c6 cdc
nhd todn hgc trong mcac X vd nude ngodi Moi nhd todn hgc quoc gia X gin
dung mgt thong di^ eho mgt nhd todn hgc nude ngodi Moi nhd todn hgc nude
ngodi gid dung mgt thong diep cho mgt nhd todn hgc cua quoc gia X Mac du
vay Cling ed it nhat mgt nhd todn hge khong nhan dugc mgt thong diep ndo cd
Chihtg minh rang ton tai mgt tap hap S gdm cdc nhd todn hgc quoc gia X vd
tap hap T cdc nhd todn hge nude ngodi thoa man cdc tinh chat sau:
1 Cdc nhd todn hge quoc gia X thugc S chi gid thu eho cdc nhd todn hgc
nude ngodi khong thugc tap hap T
2 Cdc nhd todn hgc nude ngodi thugc T chi gtti thong diep cho cdc nhd todn
hgc quoc gia X khong thugc S
Loigiai i •(-•">: ^ • r
1 Gpi A la tap hgp cac nha toan hpc cua quoc gia X va B la tap hgp cac nha
toan nuac ngoai Goi f: A -> B va g: B A la cac anh xa dinh nghia nhu sau:
f(a) la nha toan hpc nuoc ngoai nhan thong di^p ciia a va g(b) la nha toan hoc
cua quoc gia X nhan thong diep cua b Neu cac tap con S va T ton tai thi T = B\
f(S) Vay ta phai chung minh ton tai mot tap S c A sao cho A\ = g(B\f(S)) Vai
m6i tap con X c A, goi h(X) = A\g(B\f(X)) Neu X c Y thi '
v ; ; f ( x ) c f ( y ) ^B\f(Y)c=B\f(X)
; =>g(B\f(Y))cg(B\f(X))
=>A\g(B\f(X))c:A\g(B\f(Y))
^ h(X) c h(Y) G(?i M = {X c A/h(X) c X) Tap hgp M la khac rong vi da c6 A e M Mat khac
theo gia thiet g khong phai la toan anh vay ton tai HQ thugc A ma khong thugc
g(B\f(X)) vay thugc h(X) vai mgi X c A Vay mgi tap hgp trong M chua ag
Vay S = X e M n X khac rong Theo dinh nghla cua S ta c6 h(S) c S Theo tinh
chat dan difu cua h ta c6 h(h(S)) e h(S) Vay h(S) e M va S c h(S) Tong hgp
cac ket qua tren ta c6 S = h(S) (dpcm)
Sir dung mgt so tinh chat ciia anh xa
Cho A va B la hai tap hCru han
1 Neu C O mgt dan anh f : A B thi |A| < |B| ;
2 Neu C O mgt song anh f : A -> B thi | A| = B| ; '
3 Neu C O mgt toan anh f: A B thi |A| > JB
Cty TNHH MTV DWH Khang Vi?t
Trong mgt so bai toan de'm, dinh l i tren thuang dugc ap dung bang each: Ivluon de'm so phan tit cua tap hgp A , ta thiet lap mgt song anh tir A den B ma
B la mgt tap hgp da biet so phan tu
Trong bai toan bat dSng thuc to hgp, y tuong tren thuang dugc ap dung
nhu sau: Muon chung minh | A| < |B| , ta tim mgt anh xa di tir A vao B (hoac B
A) Sau do chung minh anh xa do la mgt dan anh (hoac toan anh) nhung khong phai la song anh
($di todn 4.3. ( I M O 1989) Mgt hodn vj (xi,X2, X2n> cua tap hotp (1, 2,2n) (n
Id so nguyen duang) dugc ggi la cd tinh chat P neu [xj - X j ^ i j = 1 vdi it nhat
mgt i e {1, 2 , 2 n - 1} Chicng minh rang moi n, so hodn vi ed tinh chat P Idn
han so hodn vi khong cd tinh chat P
Lai giai
(i) Cach 1 Dat A la tap tat ca hoan v i ciia (1, 2 , 2 n } ; B la tap tat ca hoan vi ciia {1, 2, 2nl c6 tinh chat P; C la tap tat ca hoan vj ciia {1, 2, 2n} khong c6 tinh chat P
Ta chia cac so 1, 2 , 2 n thanh n cap sau: (1; n +1), (2; n + 2 ) ; ( n ; 2n)
Gia su (x^; X2; ; x^^i; X;^; Xj^^^; ; X 2 n) la mgt hoan vi thugc C
Gia su la so'ciing cap vai X2n, suy ra k < 2n - 2 ,
Ta xay dung anh xa f : C B nhu sau
X = (x^, X2, X i ^ _ i , X|^, X | ^ ^ l , X 2 n ) -> Y = ( X j , X 2 , , , >-• '^k+l )
De tha'y rang f la don anh va f khong phai la toan anh
That vay, xet yo = (x^, X2 x^-i • ^n+i • ^n+i 2n )
Khi do khong ton tai XQ de f( Xg) = YQ D O do f khong phai la toan anh
Tu do ta suy dieu can chung minh
J^MnxetTa c6 |C|n|B| = 0 ma |C|+ |B| = |A| = (2n)! nen |B!
C| < n(2n - 1)! Tu do dan den each chung minh thu 2 sau:
(ii)Cach 2 Ggi Aj^ la tap tat ca hoan vi thoa man k va k + n dung canh
nhau, A la tap hgp tat ca cac hoan vi c6 h'nh chat P Suy ra A = (JAJ^
Trang 13Boi duang hoc sink gioi Todn to hap - roi rac, Nguyen Van Thong
Vi v^y A| > 2n(2n - 1)! - ^^^^^^ • 4(2n - 2)! = 2n^ (2n - 2)! > ^
(b) Khi n=0,3 (mod 4) ta c6 1 — ' — < < f(n) < 2 " - 2 L 2 J
+ 1
Loi giai (a) Gia su ton tai mpt each dat dau +, - voi n = 1, 2 (mod 4) de = 0
Khi do 1 + 2 + + n = 0 (mod 2), suy ra "^"^^ = 0 (mod 2) Ma dieu nay
xay ra khi va chi khi n = 0, 3 (mod 4) Ta c6 dieu can chiing minh
(b) Ta chung minh f(n) < 2""^
That vay, chia tat ca bieu thiic thanh 2""^ cap theo dang : (1 + 2a2 + Sa^ + +
n a ^ ; - ! + 2a2+ 3 3 3 + + nan)v6i a; e (1;-1}, i = 2,n
Neu f(n) > 2"~^ thi theo nguyen l i Dirichlet ton tai 2 bieu thuc cung nam
trong mot cap n tren, hieu cua chung bang 2 Do do chiing khong the dong thoi
bang 0 dugc (mau thuan) ^ > ^ »K '
+ 1 ,
.,»-Do do f(n) < 2"-^ = 2" - 2""^ <2"-2^^^
Ta chiing minh f(n) > -^^—'— j 4 f) ( i f
1' ^jf't.'
Xet bieu thiic E n , gpi A n la tap hop cac so xuat hien trong E n vai dau + 6 truoc
Neu En = 0 thi tong cac phan tii cua bang ^
N h u vay voi moi each chon An ta c6 the xay dung it nha't 4 tap A n + 4
Gia sir cho truac tap A n + 4 • Ta thay tap A n + 4 <5upc xay dung t u tap A n va
them diing mpt cap trong tap (n + 1; n + 2; n + 3; n + 4) va An chiia diing mpt
cap nhu the' ;
Cty TNHH MTV DWH Khang Viet
Tu do ta C O dieu can chiing minh
^di todn 4.4 (Romania TST 2002) Vai moi so nguyen ducmg, ta goi f(n) la so
each chon cac dau +, - trong bieu thiic: = ± 1 ± 2 ± ± n sao cho En = 0 Chiing
minh rang
(a) fin) = 0khin^l,2 (mod 4)
Do do ta C O the chi ra 6 truong hop khi xay dung A n + 4 • Tu do xac dinh
dugc An la duy nha't
Do do anh xa tu An den A n + 4 ^"h
Mat khac vol moi An ta xac dinh dupe duy nha't mpt , voi moi En cung xac dinh dupe duy nha't mpt An
Do do f(n + 4) > 4f(n) ta c6 f(3) = f(4) = 2 Suy ra ' + f(4k + 3) = f ( ( 4 k - l ) + 4 ) > 4 f ( 4 k - l ) > 4^f(4f-5)> > 4''f(3)
(^)
= + f(4k + 4) > f((4k - 1) + 4) > 4f(4k - 1) > 4^ f(4f - 5) > > 4'' f(4)
4^2=-_ ( V 2 p
Vi vay f(n) n) > — ' — Vay ta c6 di Vay ta c6 dieu can chiing minh
(gdi todn 4.5 Cho tap hap A = {1; 2 ; 2 n } Mot tap con C cua A duoc goi la mot
tap tot neu trong tap do so cac sochan it han so cac sole Mot tap con B cua A duoc goi la mot tap can neu trong tap do sosochdn bang sole Bat so tap tot
la T, so tap can la C Chiing minh rang: T+C< 2^""^
Loi giai
Ki hieu X = (2; 4 ; 2 n ) la tap tat ca cac so chin eiia A ; Y = {1; 3 ; 2 n - 1} la tap tat ca eac so le ciia A
Gpi D la hp tat ea cac t^p can ciia A ; E la hp tat ca tap con n phan tii ciia A
• Tinh so tap can: , * ^'
Gia sii B la mpt tap can ciia A , B 2 Ian lupt la tap cac so'chan va le ciia B,
Suy ra f(B) e E De thay f la don anh Ta chiing minh f la toan anh
That vay, gia sii M la mpt tap con c6 n phan tii ciia A
Gpi , IVI2 Ian lupt la tap cac so chan, cac so le ciia M Dat Bi = M l , B2 = Y \ M 2 , B = B^ u B2 , , ,
Trang 14Boi duang hoc sink gioi Todn to hap - red rac, Nguyen Van Thong
Suy ra B2I = |Y\M2| = |Y| - IM2I = n - IM2I = |MI| D O do B la
m p t tap can hay f la toan anh
V i vay f la song anh T u do suy ra so tap can la C^n •
fj • T i n h so'tap to't:
Gia su C la m o t tap tot ciia A
Gpi Ci, C2 Ian l u g t la tap so chan va le ciia C thi
Gpi F la hQ talt ca tap tot ciia A /' < J 4 t ' } > ^
Ta thiet lap anh xa sau
T u do ta CO bai toan sau:
^di todn 4.6 Chiing minh rang so'cac each bieu dim mot songuym duong n
thanh tong ciia cac songuyen dumg ma khong c6 sonao chia hei cho p (p la
mot so cho truac) xuai him qua mot Idn khong Ian han so each bieu dim n
thanh tong cac songuym duang ma khong eo sonao chia hei cho
L a i g i a i
" Xet bo f , 8 2 , } ^ao cho trong bo khong c6 phan t i i nao chia het cho p
m
xuat hi^n qua m g t Ian va ^ ^ B j = n
Bieu dien tat ca cac so chia het cho p d u a i dang p*" t (r > 1 ; t / p)
G i a s u { a i , 8 2 , , an} = { b i , b,,, p^i.t^; ; p'"'.tt)
Gpi M la tap tat ca cac each bieu dien n thanh tong ciia cac so nguyen
d u a n g ma khong c6 so nao chia het cho p (p la mot so cho truoc) xuat hi^n qua
m p t Ian va N la tap tat ca cac so bieu dien n thanh tong cac so nguyen d u o n g
ma khong c6 so' nao chia het cho p^
Xet anh xa sau: f : M N
{ a i , a^} h->
V- ' • ''''
bi,b2 b i , , p t i , p t i , p t i , p t 2 , p t 2 , - p t 2 , - P t t > P t f - P t t
De thay f la m p t d o n anh nen ta c6 dieu can chung m i n h
Cty TNHH MTV DWH Khang Viet
DAI s6 T 6 HOP
§1 P H f i P D E M
J\fhdn xet: M a t khau de truy cap vao m p t h | may tinh g o m sau, bay hoac tarn ky su M o i k y t u c6 the la m p t c h i i so hay m p t c h u cai M o i mat khau phai chii'a it nha't m p t chir so
H o i CO bao nhieu mat khau n h u vay? N h i r n g k y thuat can thiet de tra l o i cho cau hoi do va m p t lop rpng Ian cac bai toan dem khac se dupe gioi thieu trong chuang nay
Bai toan dem cac phan t u xuat hien rat nhieu trong toan hpc cung n h u trong tin hpc V i d u chiing ta can dem so n h i i n g ket qua thanh cong ciia cac thi nghiem va toan bp n h i r n g ket qua kha d i ciia n h i i n g t h i n g h i ^ m do de xac dinh xac sua't cac bie'n so r a i rac C h i i n g ta can tinh so cac phep toan phai lam trong mpt thuat toan de nghien ciiu dp phiic tap ciia no
Trong phan nay chiing ta se trinh bay cac p h u o n g phap d e m co ban, chiing
la nen tang cho hau n h u tat ca cac p h u o n g phap khac
1.1 Q U Y T A C C O N G : r •'r.-i Cho A, B la hai tap him han va A n B = 0 thi |A| + |B| = |A + B| ' V '
Xay d u n g song anh f : A -> (1, 2 , m }
^ ^1 = ^2 (Vi f, g deu la song anh)
Ngoai ra, h toan anh v i m p i phan t i i ciia h deu c6 anh va m p i phan t i i ciia
2 , m + n) deu c6 tao anh nen h la toan anh Vay h la song anh
h : A u B - » { 1 , 2 , m + n} la song anh hay | A U B | = |A| + |B
, 1/ '1 111 i 1
f ( x i ) = f ( x 2 ) v a g ( x i ) = g(x2)
Trang 15Boi duang hqc sink gidi Todn to hap - rbi rac, Nguyen Van Thong
1.2 QUY TAG C O N G GHO n TAP HOP;
N e u A j , A 2 , A n la cac tap h i i u han d o i m o t r o i nhau, tiic la A j n A j = 0
neu i ; ^ j , t h i A;^ ụ-.u = | A I | + | A 2 | + + A ^ ] ,
6 day JAjj la luc l u g n g (so cac phan ttr) cua tap Aj
, Chung minh Gia su Ậ-. la cac tap h u u han do i m p t r o i nhaụ Bang
q u i nap theo n, ta c h u n g m i n h rang: Â u u A n = Â + + A„
V a i n = 1, dang thuc tren hien nhien d i i n g V o i n = 2, dang thiic tren suy ra
t u d i n h nghia t o n g cua hai ban sọ Gia su dang thuc tren da d u g c c h u n g minh
cho n = k > 2 va Ai, ,A|^+i la k + 1 tap h i n i han doi mot r o i nhaụ K h i do
(Aj ụ uAi^) va Aj^^i ciing r o i nhaụ Theo gia thiet q u i nap, ta c6:
A i U u A k U A k + i = | ( A i U u A J u A k + i
+ + +
= ( A i U u A J
,j iNhdn xet guy tdc cong:
Gia su ta c6 n hanh d g n g loai t r u Ian nhau Hi, ,Hn, tiic la k h o n g the xay ra
hai hanh d g n g d o n g thoị Ta ciing gia su rang hanh d g n g H i c6 3 ; each thuc
h i f n K h i do hanh d g n g H : hoac xay ra, hoac H2 xay r a , h o a c H„ xay ra,
CO ca thay + +an each thuc hien
1.3 Q U Y T A C N H A N : «
' Cho A, B la hai tap h u u han va A n B = 0 t h i AxB = A B
C h u n g m i n h : Gia su |A| - m, |B| = k va ,
A = {ai,a2, ,â}, B = {bi,b2, ,bk}
Tich De-cac A x B g o m cac cap (aj.bj), 1< i < m, 1< j < k , c6 the viet thanh
m g t bang chir nhat c6 m d o n g k cgt n h u saụ /• 5 >i Jíi I » J
ftu A j n A j = 0 n e u i ^ j va AxB = A i u A 2 U u A n , ! i > rUJ •
Theo q u i tac cgng, ta c6: |AXB| = |A]^| + IA2I + + |An,| = m.k = |A| |B
Chung minh Ta ciing c h i i n g m i n h dang thuc tren bang q u i nap theo n V o i
dang thuc la hien nhien d u n g V a i n = 2, dang thuc suy ra t u d i n h nghia tich cua hai ban só Gia su dang thuc da dugc c h u n g m i n h cho n = k > 2 va ,Ak,Ak+i la k + 1 tap hiru han bat kỵ K h i do gia thiet q u i nap
A i X XAk XAk+i I = |(Ai X XAk)xAk+i
= |(Ai X XA |Ak+i|- |Ai| |A2| |Ak||Ak+i
J^an xet qui tdc nhdn: ; ^ ^
Gia su m g t hanh dgng H bao gom n giai doan ke tiép va dgc lap v o i nhau, trong do giai doan t h u i la hanh dgng H j Ta cGng gia su rang hanh dgng Hj c6
Bj each thuc hien K h i do hanh dgng H c6 ca thay ai,a2, ,an each thuc hien
'Bdi todn chia khda J
Co bao nhieu anh xa t u mgt tap hgp X eo k phan t u toi m g t tap Y c6 m phan t u
D o n g t h u hai f ( x i ) , f ( x 2 ) , f ( X k ) la m g t day k p h a n t u cua Ỵ N o la
mgt phan t u ciia rich De-cac Y x Y x Y x x Y = Y""
Dao lai, m g i day k phan t u cua Y la (y„^ ,y^^ <-,yak) xac d i n h m g t anh
x? f tir X t o i Y, neu ta dat f ( X i ) = y ^ j Su t u o n g u n g do g i i i a tap h g p cac anh xa
X t a i Y va rich De-cac Y"" la m g t song anh Vay so anh xa riJ X t d i Y bang
yk
= m
Jle qud:C6 bao nhieu each phan phoi k do vat vao m ngan keo ?
Chung minh M o i each phan p h o i la m g t anh xa tir tap h g p k do vat vao tap
^liVp m ngan keọ V i v$y c6 m"^ each phan phói k do vat vao m ngan keọ
29
Trang 16Boi duong hoc sinh gioi Todn to hop - rcri rac, Nguyen Van Tfiong
1.5 C A C Vf DU MINH H O A
Vi du l:Mot ban ddi c6 2 day ghe dot dien nhau, moi day gont c6 6 ghe
Nguai ta muon xep cho ngoi cho 6 hoc sinh trudng A va 6 hoc sinh truang B
vao ban noi tren Hoi c6 bao nhieu each xep cho ngoi trong moi truang hap sau:
a) Bat ki 2 hgc sinh ndo ngoi c^nh nhau hoac dot di?n nhau tht khdc
ChOng minh: | A | + | B | la s o ' t i m d u o c k h i ta de'm t r u a c he't so'tat ca cac p h a n
^ cua A r o i sau d o so tat ca cac p h a n t u B N h u n g k h i do, so p h a n t u c h u n g
cho ca h a i tap h o p A va B tuc la A n B| d u o c t i n h hai Ian D o d o
+ Jie qud:Gia six A , B la h a i tap hiJu han ne'u B c A t h i B
Vidii 2:Trong mot dethi co ba caw mot cdu veSohgc, mot cdu veGidi tich, mot cdu veHinh hgc Trong 60 tht sinh dvc thi, co 48 tht sinh gidi duoc cdu So hgc, 40 thi sinh gidi duoc cdu Gidi tich, 32 thi sinh gidi duoc cdu Hinh hgc Co 57 thi sinh gidi duoc cdu So hgc hoac Gidi tich, 50 thi sinh gidi duoc cdu Gidi tich hoac Hinh hgc, 25 thi sinh gidi duoc ca hai cdu So hgc va Hinh hgc, 15 thi sinh gidi duoc cd ba cdu Hoi co bao nhieu thi sinh khonggidi duoc cdu ndo?
L o i g i a i
K i h i f u T la tap tat ca cac t h i sinh A , B, C Ian l u g t la tap h g p cac t h i sinh
giai d u g c cau So hoc, G i a i tich, H i n h hgc Theo t i n h chat 2 ta co:
Vay CO 3 t h i s i n h k h o n g g i a i d u g c cau nao
Vidu 3:Khi dim tra kei qud hgc tap cdc mon Todn, Ly, Hoa cua mot lap co
45 hgc sinh, nguai ta nhan thay: co 19 hgc sinh khong gidi mon ndo, 18 hgc
sink gidi Todn, 17 hgc sinh gidi Ly, 13 hgc sinh gidi Hoa, 10 hgc sinh gidi hai
^on Todn vd Ly, 9 hgc sinh gidi hai mon Ly vd Hoa, 10 hgc sinh gidi hai mon T^odn vd Hoa Hdi bao nhieu hgc sinh gidi cd ba mon?
31
Trang 17Boi duang hgc sinh gidi Todn to hop - rai rac, Nguyen Van Thong
Lai giai
Ki hieu T la tap hgp hoc sinh ciia lap A, B, C Ian lugt la tap hgp cac hgc
sinh gioi Toan, Ly, Hoa cua lop do
Vi du 4: Tim hieu kei qua hoc tap d mot lap hoc, nguai ta thay:
• Han - so hgc sinh dcit diem gioi a mon Todn cung dong thai dat diem
3
• Han - so'hgc sinh dat diem gioi & mon Vat ly cUng dong thod dat diem
3
gioi d mon Van;
2 ^
« Han - so hgc sinh dat diem gioi a Van cung dong thai dat dieM giai a
mon Lich sit;
e Han - so hgc sinh dat diem gioi a mon Lich sii cung dong thai dat diem 3
gioi a mon Todn;
Chicng minh rang trong lap hgc noi tren c6 it nhat mot hgc sinh dat diem
gioi cd ban mon Todn, Vat ly, Van vd Lich SM'
(De thi MSG Quoc gia THPT Bang B-2005)
Ta giai bai toan bang phuang phap phan chung
Gia su.khong c6 hgc sinh nao dat diem gioi 6 ca bo'n mon Toan, Vat ly Van
va Lich su, khi do chi con: T n V = 0 hoac L n S = 0
Tix (1) va (2) ta gap mau thuan nen dieu gia su ban dau la sai
• Neu L n S = 0 lap luan tuong tu ciing dan den dieu mau thuan Bai toan dugc chung minh ^ ^, , ^j,,,,^^^., ,
§2 T O H O P K H O N G LAP
jVhan x^t: Gia su mgt dgi quan vgt c6 muai cau thu huan luy|n vien can
chgn nam nguai di thi dau a mgt truong khac Ngoai ra ong ta cung chuan bi
mpt danh sach c6 thu tu gom bon cau thii de tham gia bo'n tran chai don
Trong muc nay ta se nghien cuu cac phuang phap de'm so each chgn khong c6 thii tu nam cau thu de di thi dau va c6 danh sach khac nhau gom bon cau thii
tham gia tran choi don ^ Tong quat han, chung ta se trinh bay cac phuang phap de'm so each chgn
khong CO thu tu cac phan tu khac nhau va nhirng sap xep c6 thii tu cac do'i
hrgng cua mgt tap hiiij han
2.1 T6 HOP (J6 HOP KHONG LAP)
So'tap con ciia mgt tap hop c6 m phan tu
Cho t^p hgp A c6 m phan tii Ta hay xet xem tap hgp tat ca cac tap con ciia
^O/ T(A), CO bao nhieu phan tu
^<ii todn 2.1.1 So tac ca cac tap con ciia mgt tap hgp c6 m phan tu bang 2""
T(A) =2"
Chung minh:Ta dua ra hai each chung minh khac nhau
33
Trang 18Boi dtithi}; hoc sink gioi Todn to h<pp - rai rac, Ngm nong
Cdch l G o i Tg la tap hop tat ca cac tap con cua tap hop A chua phan tij
a 6 A Hien nhien moi tap con nhu the dugc hoan toan xac dinh, neu ta biet ta't
ca cac phan tu con lai cua no (tru a) Vi vay c6 bao nhieu tap con nhu the thi c6
bay nhieu tap con trong tap hop A ' = A - {a) Tap hop A ' nay c6 m - 1 phan tir,
Vi vay, neu ta gpi S^, la so tap con cua mgt tap hg-p c6 m phan tu, thi
Ggi Tg la tap hgp tat ca cac tap con ciia tap hgp A khong chua a, thi \g
bang sm-i vi cac tap con do ding la cac tap con cua tap hgp A ' = A - {a}
Vi T{A) = T,uT,
Va Tg n Tg = 0 , nen theo quy tac cgng, ta c6: T ( A )
Tudosuy ra s„ =2s„_i
= 2s m-l
A p dyng lien tiep dang thiic nay, ta dugc: S„ = 2Srn_i = 2^s^_2 = - - 2 ' " " ^ S i
Sj la tap con cua mot tap hgp c6 mot phan tu Nhung mot tap hgp c6 mot
phan tu chi c6 hai tap con la toan bg tap hgp va tap hgp rong Vay = 2 Do
d 6 : s „ =|T(A)| = 2'"
Cdch 2: Phuang phap anh xa
Cho tap hgp A c6 m phan tir Xet tap hgp Y = {0, 1) Voi moi tap con B cua
A , ta xac dinh mgt anh xa f : A -> Y nhu sau: Cho x e A , neu x e B thi ta dat f(x)
= 1, con neu x g B thi ta dat f(x) = 0
Nhu vay ling voi moi tap con B ciia A , c6 mgt anh xa f tir A toi Y
Dao lai, neu f la mgt anh xa tir A toi Y thi ung voi no c6 mgt tap con B cua
A , gom tat ca cac phan tir x e A sao cho f(x) = 1
Svr tuong ung ay giira tap hgp cac tap hgp cua tap hgp A va tap hgp cac
anh xa tir A toi X ro rang 1 - 1 Do do so tap con cua A , tuc la T { A ) , bMng so
anh xa tir tap hgp A c6 m phan tir toi tap hgp Y c6 2 phan tir Ta bie't rang so
nay la 2™ Vay: |T(A)| = 2 "
2.2 T 6 HOP:
^inh nghia 2.2.1. Mgt tap con k phan tir cua mgt tap hgp m phan tir dugc ggi
la mgt to hgp chap k cua m phan tir
So' to hgp chap k cua m phan tir
Muon dung mgt tap con k phan tir cua tap hgp A, ta c6 the ghep them vao
mpt t?P ^' ^ P^^^ ^ "^9t trong m - ( k - l ) = m - k + l phan tir khong
thamduvaono
Vi CO tap con k - 1 phan tir va ta c6 the bo sung moi tap con ay thanh
mgt tap con k phan tir theo m - k + 1 each, nen lam nhu vay ta thu dugc ( m - k + l)Cm"^ tap con k phan tir cua A Nhung khong phai tat ca cac tap con nay deu khac nhau, vi ta c6 the dung moi tap con k phan tir theo k each, cu the
la lay moi mgt trong k phan tir cua no ghep them vao k - 1 phan tir con lai Vi vay so (m - k + l)ct^~^ vira tim dugc d tren gap k Ian so' cj^ cac tap con k phan tir cua A Do do ta c6 dang thuc: (m - k + l)cj^~^ = kCJ^ riji) i
•linh mgt giao diem (do la giao diem cac duong chte cua t i i giac xac djnh boi cac dinh ay) Vi vay so tat ca cac giao diem bang so to hgp chap 4 cua n dinh
n ( n - l ) ( n - 2 ) ( n - 3 ) n ( n - l ) ( n - 2 ) ( n - 3 ) ; f
" 1.2.3.4 24
Vi du 2: Co bao nhieu cdch phan phot 5 do vat cho 3 nguai, sao cho moi
^Sum deu nhan dugc it nhai mot do vat? , M
Trang 19Boi duang hgc sitth gidi Todn to hap - riri rac, Nguyen Van Thong
L 6 i giai
N e u khong ke den t i n h giao hoan ciia phep cpng, t h i so' 5 c6 the phan ticli
thanh m p t tong cua 3 so' ty; nhien khac 0 theo cac kieu khac nhau sau day:
5 = 1 + 1 + 3 = 1 + 2 + 2 = 1 + 3 + 1 = 2 + 1 + 2 = 2 + 2 + 1 = 3 + 1 + 1
Lfng v d i cac kieu phan tich do, c6 cac kieu phan pho'i 5 d o vat cho 3 nguoj
A, B, C sao cho m o i n g u o i deu nhan dupe it nhat m p t d o vat n h u sau:
Theo kieu I , A nhan dupe 1 d o v^t trong so'5 d o vat, vay A c6 C3 = 5 each
chpn B nhan dupe 1 d o vat trong so'4 d o v|t con lai, vay B c6 the chpn theo
C4 = 4 each C o n C t h i nhan 3 d o vat con lai theo d u n g 1 each Theo q u y tac
nhan, so each phan phoi theo kieu I la C5C4.I = 5.4.1 = 20 each
Lap luan t u o n g t y , ta se tha'y rang, phan p h o i theo kieu I I c6
C^C^.1 = 5.6.1 = 30 each; theo kieu I I I c6 C^cil = 5.4.1 = 20 theo kieu I V c6
C^C^.1 = 10.3.1 = 30 each, theo kieu V c6 C^C^.l = 10.3.1 = 3 0 , theo kieu V I c6
Vay cac so C ^ + i = la Ion nhat ^ ^ ,
Vi du 4: Cho da gidc dm AiA2 A2n (n^ N va n>2) noi tiep trong duang trdn (O) Biei rang so"tarn gidc co dinh Id 3 trong 2n dinh Ai,A2, ,A2n nhieu gap 20 Idn so hinh chit nhat co cdc dinh la 4 trong 2n dinh A^, A2, A2n Tim n
c o n each ^ .) ,, , „ , , , - • ,'.:„
Chpn 1 d i n h N bat k i trong cac d i n h con lai co 2n - 2 each
Luon luon t i m dupe N ' doi x u n g qua tam O de M N M ' N ' la h i n h c h i i nhat
N h u n g d o m o i h i n h chxx nhat M N M ' N ' n h u vay b i de'm t r i i n g lai 4 Ian nen so
u< u , - , , , ,^ n { 2 n - 2 ) n { n - l ) hinh c h u nhat tao thanh la: -^^ = — -
4 2 / ,(
Do so' tam giac nhieu gap 20 Ian so' hinh chCr nhat, nen:
o ( 2 n - l ) ( 2 n - 2 ) = 3 0 { n - l ) ( d o n > 2 )
o ( 2 n - l ) = 1 5 < ^ n = 8
Vi du 5: Mot tap theco 14 ngubi gom 6 nam va 8 nii trong do co An va Binh
^guai ta muon chon 1 to cong tdc gom 6 ngubi Tim so cdch chon trong moi truang hap sau:
«) Trong tophdi co mat cd nam Ian nit
b') Trong tophdi co 1 to truang, 5 to vim, han nita An vd Binh khongdong
Trang 20Boi duang hpc sink gioi Todn tohqp - rcri rac, Nguyen Van Thdng
So'each chon 6 nguoi toan nam: Cg = 1
So' each chon 6 nguoi toan nu: Cg 8!
So'each chon An lam to vien va khong c6 Binh: IZ.Cn - 12.-^^ = 3960
Vay so'each chon CO An ma khong CO Binh: + IZC^^ = 4752
Tuong tu so'each chpn c6 Binh ma khong c6 An cung la: + 12Cji = 4752
So each chon khong eo An Ian Binh: 12C^i = 1 2 - ^ = 5544
Do do yeu cau bai toan: 2(0^2 + UC^i) + UC^^ = 2(4752) + 5544 = 15048
Cdch2: x ; , ,//
Chon tuy y 6 trong 14 hoc sinh eo: each
Chon An va Binh roi chon them 4 hoc sinh trong 12 hoc sinh eon lai c6: C^2 each
Vay so' each chon 6 hoe sinh do An va Binh khong dong thai c6 mat:
Vol 6 hoc sinh da chpn xong c6 6 each ehpn ra to truong
Vay so each ehpn thoa yeu cau de toan la: sicf^ -^12) = 15048 each
2,4 Y NGHlA HINH HOC CUA = cf^.^j.^ •
Xet mot mang duang hinh chir nhat kich thuae m x n, tao thanh bai nhiing
hinh vuong, ngan each nhau boi n - 1 duong ngang va m - 1 duong dpc nhu
tren hinh ve Ta hay tim so duong ngan nhat tren mang duang do de di tu goc
dual ben trai diem (0, 0) tai goc tren ben phai (diem (m, n))
( 0 , n )
-(m, n)
(m, 0) X
•58
Cty TNHH MTV DWH Khang Viet
Moi duang ngan nhat di tu diem (0, 0) den diem (m, n) deu gom m + n doan thang, trong do c6 m doan ngang, n doan doe Cac duang ay chi khae nhau boi thii tu ke'tie'p eiia eac doan ngang va cac doan dpc Vi vay so tat ea cac duong jigan nhat tu diem (0, 0) tai diem (m, n) bang so each ehpn n doan dpc tu m + n do^n dpc ngang, hie la bang Cj^+n
Nhu vay, so Cj^ = c|^^_^)^,^ c6 y nghia hinh hpc nhu sau: Do la duang ngan
nhat tir diem (a 0) tai diem (m - k, k) ^ ' M " ' '
Chii y: Ta eung eo the xet so' each ehpn m doan ngang thay cho n doan dpc Khi do so duong ngan nhat tu diem (0, 0) toi diem (m, n) se la Cj^^^ Nhu vay
ta da chung minh dupe bang hinh hpc dang thuc C^^^ = Cj^+n
2.5 MOT S 6 TINH CHAT QUAN TRONG CUA CAC S6
Zi
2.5.i.Ne'u 0 < k < m t h i (2) Cong thuc nay dupe gpi la quy tac do'i xung
Chicng minh:
Cdch 1: Theo y nghia hinh hpc cua so' cJ^=cL_|^wk, ta da thay rang
C(m-k)+k = • chinh la cong thuc (2)
Cdch2:Neu dimg eong thuc (1) thi ta eo ngay
{ m - k ) ! ( m - ( m - k ) ! ) ( m - k ) ! k !
V nghia tap hop eua eong thuc (2) nhu sau: Gia su tap hop A c6 m phan tu Ne'u B mpt tap con k phan tu eiia A, phan bu B eua B trong A la mot tap con (m - k) phan t u cua A N h u vay giira tap hop cac tap con k phan tu ciia A va tap cac con (m - k) phan tu eiia A c6 mot tuong ung 1 - 1 Do do so' tap eon k phan tu bang so'tap con m - k phan tu Dieu nay dupe dien ta boi eong thiie (2)
•2.5.2 Ne'u m va k la nhiing so tu nhien sao cho 1 < k < m - 1 thi ta c6
Trang 21Boi dudmg hqc sink gioi Toan to hap - rcri rac, Nguyen Van Thong
Cdch 2;Cho tap hop A c6 m phan tu Xet mot phan t u a € A Khi do cac tap
con k phan tu cua A chia thanh hai lop: lop thu nhat gom nhiing tap con
khong chua a, lop thii hai gom nhiing tap con chua a Neu mot tap con k phan
tu khong chiia a thi no la mot tap con k phan tu cua tap hop A' = A - {a} Vi
vay so' tap con k phan t u ciia lop nhat bang so tap con k phan t u cua tap hgp m
- 1 phan t u A' So' do la CJ^_i Ta hay tim so' tap con k phan tu ciia lop thu hai
Tat ca cac tap con nay deu chua a Neu ta bo phan tu a di, thi ta duoc nhiing
tap con k - 1 phan tu, khong chua a va gom toan nhirng phan tu cua A' Vay so
tap con cua lop thii hai bang so tap con k - 1 phan tu ciia tap hop A' c6 m - 1
phan tu, tuc la c]^"l\
Vi moi tap con k phan t u hoac chua a, hoac khong chua a nen no thupc hoac
lop thu nhat, hoac lop thu hai Vay theo qui tac cpng, ta c6:
Ta CO the chia cac duong do thanh hai lop khong giao nhau: lop thu nhat
gom cac duong di qua diem A^ (k - 1, m - k), so phan tu cua lop nay la
^('k-l)+(m-k) - ^ m - l ' ^^P ^hu hai gom cac duong di qua diem A2 (k, m - k - 1),
so'phan tu cua lop nay la c{^^^j^_|^_^j = c[^_i Theo quy tac cpng, ta c6:
pk _ / - k - l pk
Cty TNHH MTV DWH Khang Viet
Vi da 6:Chiing minh hang dang thiicCl, + C^^ + + + c;;^ = 2"^
Vay ta co: C + C^, + C^^ + + = 2^ (dpcm)
Vi du 7: Chung minh hhng ddng thiic =('^lf + (cj, f + + (qf
•; Loi giai r s^'S' •
So duong ngan nhat di tir diem O (0, 0) toi diem A (n, n) bang Moi
duong nhu the deu di qua mot va chi mot diem Ak (k, n - k) nam tren duong cheo BD ciia hinh vuong ABOD ,
So'duong di tir diem O toi diem Ak (k, n - k) bang c{^^^^_,^j = Mot duong
ngan nhat di t u Ak (k, n - k) toi A (n, n) gom c6 n - k doan ngang va n - (n - k)
= k doan dpc Vay so' duong di tu Ak toi A bang c}^^^^_|jj = Theo quy tac
nhan, so duong ngan nhat di t u O toi A qua Ak bang CJ^.C^ ={Cn] • Theo quy
tac cpng, so'duong ngan nhat di t u O toi A bang tong so'duong di tu O qua Ak
(k = 0, 1, n) (vi mot duong di qua mpt trong cac diem do thi khong di qua cac diem khac)
Vi diJL 8: Cho p la mot songuyen tole, n e N* (n > p ) Tim so cac tap con A
cua tap {1, 2 , n ] co tinh chat sau ^ j j , 1 , 1) A chiia diing p phan tic
2) Tong cac phan tit cua A chia he't cho p
Trang 22Boi dudng hoc sink gioi Todn to hap - red rac, Nguyen Van Thong
Xet da thuc p ( x ) = xP-^ + + + X + 1 , (1)
' D a thuc nay c6 p - 1 nghiem phuc phan biet Goi a la nghiem bat ki cua p(x)
Vi a , , a P ' ^ la p - 1 nghiem phan bi^t ciia P ( x ) va = 1
: O do so'ni la so'cac tap A e H ma S ( A ) = l(modp)
Mat khac, gia sir n = kp + r (0 < r < p-1)
i=0
Tic (4) suy ra a la mot nghiem cua R Vi degP = degQ va a la 1 nghifm bat
ky ciia P nen P va Q chi sai khac nhau hang so' nhan
P
Vidu 9:Cho bang vuong ktch thuoc: ^n^ + n + l j x ^ n ^ + n + l j Nguoi ta to den S 6 vuong cua bang sao cho khong c6 4 6 to den nao tao thanh 4 goc cua mot hinh chiinhat c6 canh song song vai canh cua bang vuong Chimgminhrang: S<{r\ \)><{n^+ +
i=i 2
So cac cap 6 c6 the cung hang bang: C^2 ^
Do khong c6 4 6 to den tao thanh 4 goc cua mot hinh chii nhat c6 canh song song voi canh bang vuong nen trong A cap 6 to den tren khi chieu xuong cimg mot hang ngang thi khong c6 2 cap nao triing nhau
,av ("'-^"F^"^^)>A^"'g^^a,(a,-l)
Do vay
i=l n^+n+1 n^+n+1
c > ( n 2 + n n^ + n + l > ^X^i'l)- ^ a ^ - S
i=l i=l Theo bat dang thuc Cauchy-Schwarz thi
S2 = n +n+l i=l
Trang 23Boi dumtg hoc sink gidi Todn tohgrp - rat rac, Nguyen Van Thong
, § 3 H O A N V I , U v , „
N h a n xet: H o a n v i va c h i n h h g p ; v ; E r , , - r ; ^ , i - o ' v / r - i b ftl
H o a n v i cua m o t tap cac doi t u g n g khac nhau la m o t each sap xep c6 t h i i t u
cac doi t u g n g nay Chung ta ciing quan tam toi viec sap xep c6 t h u t u m g t so
phan t u cua m o t tap hop M g t each sap xep c6 t h u t u k phan t u eiia m g t tap n
phan t u dugc ggi la m g t chinh hgp chap k cua n phan t u
K h i k = n, chinh hgp chap k ciia n phan t u chinh la hoan v i cua n phan t u
3 , 1 H 0 A N V ! K H 6 N G L A P :
Jiodnvi3.1.1: ' • ' ' ' > /i >
N h u n g tap h g p sap t h u t u khac nhau, ma chi khac nhau b o i t h u t u cac phan
t u (tue la dugc tao nen t u cixng mgt tap hgp) dugc ggi la n h u n g hoan v j cua
tap hgp do
^inh 1(3.1.2:
Ta CO
Chung minh:
Cdch l:Ta se Ian l u g t chgn cac phan t u cua tap hgp A va sap chiing vao m
v i t r i theo m g t t h u t u xac dinh
6 v i t r i t h u nhat ta c6 the dat bat k i phan t u nao trong m phan t u eiia A
(buoc 1: m each chgn) Sau k h i da dat phan t u t h u nhat, 6 v i t r i t h u hai ta c6 the
dat bat k i phan t u nao trong m - 1 phan t u con lai (buoc 2: m - 1 each chgn), va
tiep tuc l a m n h u the eho toi v i t r i t h i i m (buoc m : 1 each chgn) Theo q u y tac
nhan, so each sap t h u t u cac phan t u cua tap hgp A la
m ( m - l ) ( m - 2 ) 2 1 = m! , , J 7i.Kij gnisb iMo'^^/lt
Cdch 2:Ta chgn m g t phan t u a nao do cua A, va xet tat ca cac hoan v i cua A
trong d o a chiem v i t r i t h u nhat H i e n nhien so hoan v i ay bang so hoan v j eiia
m - 1 phan t u eiia tap hgp A - {a} V i vay so hoan v j ciia A trong do a chiem v i
tri t h u nhat bang Pm - 1 Ta k i hieu M la tap hgp tat ca cac hoan v i cua tap hgp
A, Ma la tap hgp tat ca cac hoan v i ciia A trong d o a chiem v i t r i t h u nhat, ne'u
A = {ai,a2, ,an,} t h i ta eo t , ; - n j U - i f • r J • + i'i" i/ '
Ne'u f la m g t song anh hi' A len A thi f(ai),f(a2), ,f(an,) la m g t hoan v i ciia
A v i : d o f la d o n anh, ta c6 f(ai);if(aj)(i9tj) va v i f la toan anh nen {f (ai),f(a2) f ( a ^ ) } = A • •
Dao lai eho m g t hoan v j a i a A :(ai^ a^^ a^^) Ne'u dat f(ak)=ai|^ (k = l m) thi f la m g t anh xa A n h xa nay la d o n anh v i ne'u a^^ =A 3] t h i
f{3k) = ^ik "^^h '^'^(^l)- N ° ^^^S l3 toan anh v i phan t u ajj^ (k = l, m) la anh
a i a phan t u ak Vay f la song anh
Sv t u a n g u n g giiia tap h g p cac song anh t u A len chinh n o va tap hgp cac hoan v i a i a A ro rang la 1-1 Vay so song anh tir A len chinh no bang P^ = m !
Vi di/ 1:C6 bao nhieu cdch sap thit tu t^p h<?p {1, 2 , 2 n } sac cho mot so
Chan deu c6 sohi^u chan?
Loigiii '
Trong tap h g p {1, 2, 2nl c6 n so chan va n so le Ta c6 the dat cac so chan vao cac v i t r i v o i so chan theo n ! each V o i m o i each dat n h u the'tuong u n g n ! each dat cac so' le vao cac v i t r i c6 so h i f u le V i vay theo quy tac nhan, c6 ca thay n!xn!=:(n!)^ each sap t h u t u tap hgp (1, 2, 2n} thoa man yeu cau a i a dau bai
Vi dii 2: Co bao nhieu hoan vi cua n phan tit, trong do 2 phan tic da cho
khdng diing canh nhau?
h i o n g l i n g (n - 2)! hoan v i a i a cac phan t i i khac Do d o so' hoan v i trong do a
va b d u n g c^nh nhau bang 2(n - 1) x (n - 2)! = 2(n - 1)! V i vay so hoan v i phai timbang
n ! - 2 ( n - l ) ! = ( n - l ) ! ( n - 2 )
45
Trang 24Boi dumtg hgc sink gidi Todn tohqp - roi rac, Nguyen V&n Thong
Vi du 3: Cho k chU hoa, m nguyen am, n phu dm (cd tat cdk + m+n chu) Tic
cdc chit do CO the lap duac baa nhieu "til" (mgt ti( la mot day chit cdi viet lien
tiep, khong nhat thiei phdi c6 nghia) sao cho trong moi tie do a vi tri dau phdi
la chic hoa, con ve cdc chic khdc phdi c6 r nguyen dm vd s phu dm trong so cdc
chicdd cho?
Loigiai
Truac het ta chon mot chu hoa Co k each chon nhu the Sau do tu" m
nguyen am ta chon ra r chu, dieu nay c6 the thuc hien theo each Cuo'i
cung tir n phu am, ta chon ra s chu, dieu nay c6 the thuc hien theo each
Sau khi da chon nhu tren, ta dat chir hoa 6 v{ tri thu nhat, roi dat r + s chu con
lai, dieu nay c6 the thuc hien theo (r + s)! each Nhu vay theo quy tac nhan ta
lap duQC tat ea kC[^C^ (r + s)! tu thoa man cac yeu cau ciia dau bai
Vi du 4: Cho m chit hoa A, B,Kvdn chic thuang a, b, m, I Tic cdc chic so
do CO the lap duac bao nhieu hodn vi sao cho moi hodn vi deu bat dau bang
mot cha hoa vd tan ciing hang mot chic thuang?
Loi giai
Lay mot cap chu: mot hoa mot thuong, chang han (A, a) Dat cac chu ay vao
vi tri cuo'i Roi dat cac chu con lai theo du moi each vao gi&a hai chii' da lay, ta
tim duoe (m + n - 2)! Hoan vi thoa man dieu kien: bat dau bang A va tan cung
bang a Vi c6 the chon mn cap chu nhu cap (A, a), nen theo quy tac nhan, eo ca
thay m.n.(m + n - 2)! hoan vi phai tim
Vidu 5:Cho tap S = {1, 2,n} vain>l vaifld mgthodn vi cua tap S Phdn
tic i cua S duac goi Id mot diem codinh neiifd) = i
Ggi Pn (k) Id so hodn vi cua tap S c6 dung k diem codinh Hay chtmg minh
k-O
Loi giai 1) Vi tong tat ca cac hoan vi eiia n phan tu eo 0,1, 2 , n diem codinh bang
tat ca cac hoan vi c6 the ciia n phan tu, hie bang n! nen eo dSng thuc
k=0 ,
' 2) Ta ehiing minh rang : Vk(l < k < n) deu c6
kF^(k) = n.F^_i(k-l) (2)
Diing (f, i) de ky hi^u cap gom hoan vj f tuy y cua n phan tu vol k diem co
dinh va i la diem tiay y trong k diem co'dinh do (tue f(i) = i)
Be
Cty TNHH MTV DWH Khan^ viet
Thuanhan Po(0) = l
£)e ly giai quan h? (2) ta hay tinh so N cac cap (f, i) bang hai each:
Mpt mat, i chay qua k diem co dinh da xae djnh, nen moi hoan vi tt-ong p„(k) hoan vi do co mat trong k cap (f, i) Boi vay N = k Pn(k)
Mat khac neu f(i) = i, thi tren tap gom n - 1 phan tu con lai (tiie cae phan tu Idiac i) hoan vi f eo k -1 diem eo dinh, nen moi mot trong n phan tu i eo mat trong p^_^ (k -1) cap Do do N = n.Pn_i (k -1), nen dang thuc (2) duoe chung minh
Tinh tong cac dang thuc d (2) theo k = 1, 2, n va dua vao dang thiic (1) bang each thay n bang n - 1 ta eo: : ,|
|;kP„(k)=XnP„_i(k-l) = n ( n - l ) ! = n!
k-O k-1
§ 4 C H I N H H O P ;
4 1 TAP C O N SAP THU" TLJ C U A M O T TAP H O P : ' ' ^ '
Bay gio ta xet tap con sap thu tu cua mot tap hop m phan tu A Ta xem tap hg-p A nhu khong sap thu tu Vi vay moi tap con cua no eo the duge sap thu tu theo bat ky each nao
So tat ca cac tap con k phan tu bang CJ^
Moi tap eon do co the duoe sap thu tu theo k! each
Bang each ay ta tim duac tat ca cac tap con sap thu tu co k phan tu ciia tap hgp A
Vi vay so'tap con sap thii tu co k phan hi ciia tap hgp A bang k!Cj^ So'nay duge ki hi?u la A!1 Ta CO: A!1 = ki- rn! m!
•k!(m-k)! (m-k)! ' Nhu vay ta da chiing minh dugc dinh ly sau:
4.2 C H I N H H O P :
Cae tap con sap thii tu co k phan tii ciia mot tap hop eo m phan tii dugc ggi
la cac ehinh hgp chap k ciia m phan tii do
Cac ehinh hgp chap k ciia m phan tii la khae nhau neu chiing eo nhiing phan tii khac nhau, hoac neu eo nhiing phan tii nhu nhau, thi thii tu cac phan
Trang 25Boi duang hoc sink gioi Todn to hap - rcri rac, Nguyen Van Thong
So'chinh hap khac nhau chap k cua m phan tit bang
^di todn chia kkoa 3: Tim so'dan dnh tic mgt tap Acok phan tit den mgt tap
B com phan tii (k < m)
Lai giai
Gia su A = {ai,a2, ,ak} va f la mpt don anh tu A tai B f anh xa cac phan tu
ai,a2, ,ak cua A theo thu tu vao cac phan tu f(ai),f(a2), ,f(aij) cua B Vi f la
dan anh nen aj ^^aj keo theo f(aj)7if(aj), (i 9^ j ; i, j = \ k) Dgt f(ai) = bj eB
thi {bi,b2,-,bk} la mpt chinh hgp chap k ciia m phan tu cua B Vay mpi dan
anh f tu A tai B deu xac dinh mpt chinh hop chap k ciia m phan tu cua B
Dao lai, gia su {b^,b2, ,bk} la mpt chinh hop chap k ciia m phan tu ciia B
Neu ta dat f(ai) = bj(i = l, ,k) thi f ro rang la mpt anh x? tu A tai B Anh xa
nay la don anh vi neu ajT^aj thi f(ai) = bj ^tbj = f(aj)
Nhu vay so' don anh tu tap hop A c6 k phan tu toi tap hpp B c6 m phan tu
bang so'chinh hpp chap k ciia m phan tu, tiic la bSng AJ^ (k < m)
Vi du 2: Trong mat phang cho n diem, trong do khong c6 3 diem ndo thdng
hang Co the dung dugc bao nhieu duanggap khiic ha hoac kin c6 k canh vd cd
dinh tai n diem dd cho?
Lai giai
Trubng hgtp 1 Duong gap khuc ho. Ne'u mpt duang gap khiic ho c6 k c^inh thi
so'dinh cua no bang k + 1 Vi vay ta phai CO n > k + 1 ^ , ,
48
Cty TNHH MTV DVVH Khang Vict
Mpi chinh hpp chap k + 1 ciia n diem deu xac dinh mpt duong gap khiic ho
CO k canh trong mat phSng Hai chinh hop khac nhau ciia ciing k + 1 diem, noi chung, xac dinh hai duang gap khiic khac nhau (nhu tren hinh ve)
Song vi cimg mpt duang gap khiic c6 the dupe cho theo hai each, ling v6i hai chieu di tren duong gap khiic do, nen trong tat ca cac chinh hpp chap k + 1 ciia n diem, moi duang gap khiic xac dinh bai mpt chinh hpp nhu the^ dupe tinh 2 Ian ,
Do do so'duang gap khiic ho phai tim la -AJ^^^ - i l i i l — — k )
Trubnghap 2: Duong gap khiic kin , j <
Moi chinh hpp chap k ciia n diem xac dinh mpt duang gap khiic kin
Tir mpt chinh hop chap k ciia n diem, bang each hoan vi vong quanh, ta dupe k chinh hpp khac nhau Tat ca cac chinh hop nay deu xac d}nh cixng mpt 'lu'ang gap khiic kin
Mat khac, cijng tu chinh hpp chap k ciia n diem tren day, neu ta hoan vj vong quanh theo chieu ngupe lai, ta eung dupe k chinh hpp khac nhau nhung '^ac dinh ciing mpt duong ga'p khiic ay
Nhu vay trong tat ca cac chinh hpp chap k ciia n diem, moi duang gap khiic kin xac dinh boi mpt chinh hpp nhu the, dupe tinh k + k = 2k Ian
Do do so duong gap khiic kin phai tim la A!^^^ = jO
Trang 26Boi duang hoc sink gioi Todn tohof! - n'ri rac, Nguyen Van Tlidng
C H I N H H O P V A T O H O P S U Y R O N G
J^lhan xit: Trong nhieu bai toan dem, cac phan tu c6 the six dung lap lai Vi
du, cac chu cai hoac cac chii so dugc diing nhieu Ian trong mot bien dang ky
xe Khi mua mot ta qua tang, moi loai c6 the lay dugc nhieu Ian Dieu nay
khong gio'ng v6i bai toan dem da neu tren, trong do chiing ta chi nghien ciiu
cac chinh hgp va so' hgp ma moi do'i tugng chi diing nhieu nhat 1 Ian Trong
myc nay chiing ta se trinh bay each giai cac bai toan dem khi cac phan tu c6 the
dugc dung nhieu Ian
Cung nhu vay, mot so' bai toan dem c6 chua cac phan tu giong nhau khong
phan biet dugc Vi du, de dem so each khac nhau ma cac chvt cai cua tu SUCCESS
CO the sap xe'p lai, can phai xem xet toi viec sap xep cac chii' cai gio'ng nhau
Dieu nay tuong phan voi cac bai toan dem dugc xet 6 tren, trong do ta't ca
cac phan t u la phan biet Trong phan nay chung ta se trinh bay each giai cac bai
toan dem trong do mot so phan tu la khong phan biet dugc
C H I N H H O P C C L A P
5.1.0INH NGHIA:
Cho mgt tap hgp A c6 m phan tu Ta riit ra t u A mot phan t u bat ky, kf hieu
no la B j roi tra lai no vao tap hgp A Ta lai rut ra t u A mgt phan tu, ki hieu no
la 83 CO the lai chinh la phan tu thu nhat) roi tra lai no vao tap hgp A Tiep
tuc thao tac nay k Ian (k khong nhat thie't < m), ta tim dugc mgt day
(ai,a2, ,ai^) gom k phan t u (c6 the triing nhau) eiia A Mgt day nhu the'ggi la
mgt chinh hgp c6 lap chap k cua m phan tu da cho
Tap hgp ta't ca cac chinh hgp c6 lap chap k lap nen tu cac phan tu cua mgt
tap hgp A c6 m phan t u chinh la tap hgp cac bg {a-^,^2,->^^) voi 3 ; G A Vay
do la tich De-cac
A X A X X A = A ' '
k Ian
Tu do suy ra ngay djnh l i sau:
^inh li 5.2.1:So chinh hap c6 lap chap k cua m phan tie, ki hieu la , dugc tinh
theo cong thuc
A"^ - A-^ = m'<
Vi du 1:C6 n quyen sdch, moi quyen c6 p ban Ngudi ta ddnh son quyen
sack do theo mot thti tu nao do, va lay kj^ ban quyen thii nhat, k 2 ban quyen
50
Cty TNHHMTV DWH KhattgViet
fHii•"' ^" quyen thti n, (a < < p, i = 1, n; neu kj = 0 thi dieu do c6
fPghi(^ khong lay quyen thii i) Hoi c6 bao nhieu each chon sdch nhu the7
De thanh lap so'x ta chi can lay mgt soy nao do roi them 2 vao cuo'i Boi vay
so X = abc2 bang so y = abc va bang A4 = 4^ = 256
Chang han: 1112,1122,1132,1152,5542, 5552
Vidu 4: Co the lap dugc bao nhieu bien soxe v&i hai chit cdi ddu thugc tap {A, B, C, D, E}, tiep theo la mgt songuyen duang gom 5 chii cdi chia hei cho 5
Lai giai
Gia su mgt bien so xe nao do co dang XYabcdf
Vi X, Y CO the trimg nhau nen XY la chinh hgp lap chap 2 ciia 5 phan t u A,
^' C D, E, F, nen so each chgn XY bang A^ = 5^ = 25
Do a 0, nen c6 9 each chgn a la cac chir so thap phan khac 0
Vi abcdf-5 <=> f = 0 hoac f = 5 nen co hai each chgn chii so'f
Do b, e, d CO the triing nhau, nen moi so bed la mgt chinh hgp chap 3 ciia
^_^chir so 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Boi vay so each chgn so bed la
^10=10^=1000
Vgy so bien so xe co the thanh lap theo yeu cau la: 25 x 2 x 9 x 1000 = 45000
51
Trang 27Boi duang hgc sink gioi Todtt tohfrp - rax rac, Ngiiyen Van Thong
§ 6 H O A N V I C 6 L A P
Hoan vi c6 lap la mpt chinh h^p c6 lap, trong do c6 ke den so Ian lap lai cua
moiphantii' , s
So phan ho^ch cua mpt t|p hgp h m i h^n
<Bdi todn chia khoa 4: Cho mot tap hap A c6 m phan tii va nhimg so
ki,k2, ,ks sao cho k ^ + k2+ + ks = m Co bao nhieu each phan tap hop A
thanh hqtp red rac cua s tap con Bi,B2, ,Bs, tUc la A = B i U B 2 U u B 5 ,
B j O B j ;t0(i9tj,i,j = l, ,n), vai dieu kien |Bi| = ki;|B2| = k2, ,|Bs| = ks ?
MQt each phan tap hop A nhu the dugc goi la mgt phan hoach cua A
Ta CO the thuc hi?n cac phan hoach mo ta tren day cua tap hop A thanh s
tap con Bi,B2'-'^s ^^^u sau: Ta lay mot tap con B^ chiia ki phan t u cua tap
hop A (dieu nay c6 the thuc hi§n theo each), trong m - k^ phan t u con lai,
ta lay mpt tap con B2 chua k2 phan t u (dieu nay c6 the thuc hi?n theo cj^^
each) v.v Theo quy tac nhan, so tat ca cac each chpn cac tap eon B^-.-.B^ la
Sdi todn chia khoa 5: Gid su: c6 m chu cdi, trong do c6 k^ chu a^Mz chu
32, ,kg chii as (k^ +k2 + + ks = m ) Tii cac chit cdi do, co the lap dugc bao
nhieu "tii" khde nhau (c6 nghia hoac khong) moi titgom m chu cdi da cho?
Lai giai ' •
Ta danh so'vj tri cac chu cai trong moi t u boi cac so' 1, 2, m Goi B^ cac so
hieu ciia cac vi tri tai do c6 chii ap , tap hpp B^ cac so'hi^u ciia cac vi tri tai
do CO chir as Do do, so tu khac nhau lap nen t u cac chi> cai da cho bang so
each ma ta c6 the bieu dien tap hgp A = (1, 2, m) duoi dang hop roi rac cua
m' cac tap con B i , B 2 B ,, tiic la bang: P^(ki,k2, ks)= ^^^^^ ' ^ ,
6 1 H O A N V ! C 6 L A P :
^inh nghia 6.1.1: Cho s phan tu khac nhau, danh so t u 1 tai s Mot chinh
hpp CO lap chap m cua s phan t u da cho, trong do c6 k^ phan t u thu nhat,
Cty TNHH MTV DWH Khang Vi?t
phan tu thu hai, phan tu thu s, dupe gpi la mpt hoan vi c6 l^p cap m
^=ki+k2+ + k s ) va kieu (k^,k2, ,k5)ciia s phan tu da cho
Nho khai ni^m hoan vj c6 lap, cap m, kieu (ki,k2, ,ks), can cu vao each (Joan nhan tren day cua cac he so da thuc, ta c6 the phat bieu lai dinh li da thiet
0 tren, duoi dang sau:
^inh li 6.1.2:So hoan vi c6 lap, cap m, kieu (ki,k2, ,ks) cua s phan tie da cho,
bSng: P(ki,k2, k,)=^^^^^- (1) J
Sau day la mpt each chung minh khac dinh l i quan trpng nay
Chung minh:Xet mpt hoan vj c6 lap cap m, kieu (ki,k2, ks) Ne'u ta thay
the tat ca cac phan tu giong nhau bang nhiJng phan tu khac nhau, thi so hoan
vi khac nhau cua m phan tu giong nhau, ma ta c6 the lap dupe tu hoan vj c6 lap dang xet, theo quy tac nhan, bang ki!k2! ks!. Lam nhu vay cho moi hoan
vj CO lap cap m, kieu (kj,k2, k5), ta se tim dupe tat ca m! hoan vi cua m phan
tu khac nhau Do do ta c6 dSng thuc: P(ki ,k2, ,ks ).ki !k2 i.-k^! = m!
So'nay chinh la Vay so hoan vi c6 lap cm hai phan tit cap nt, kieu (k, m-k)
bang so'tohgp chap k cua m phan tie • - J-:
I ^ ( k , m - k ) = ci^
Ta CO the chung minh dSng thiie nay khong dua vao cong thuc (1) That
v|iy, mpt hoan vj c6 lap eiia hai phan t u a va b, cap m, kieu (k, m - k ) dupe tao thanh boi k phan t u a va m - k phan tu b No dupe hoan toan xae dinh boi each chpn vi tri ciia phan t u a Vi tong so' vi tri bang k + (m - k) = m, va phan tu a chie'm k vi tri, nen c6 the chpn cac vi tri do theo CJ^ each
Vidu 1:C6 bao nhieu each dat 2 den xanh va 4 den do thanh mot hang?
Lai giai Moi each dat 2 den xanh va 4 den do thanh mpt hang la mpt hoan vi co lap cS'p 2 + 4 = 6 va kieu (2, 4) Vay so each dat 2 den xanh va 4 den do thanh mpt hang bang so'hoan vi co lap cua 2 phan tu cap 6 kieu (2, 4), tuc la
(2,4) = c i = ^ = 15 each
53
Trang 28Bdi ducmg hpc sink gioi Todn t6h(?p - rai r^c, Nguyen Van Thong
Vi du 2:Cho p + q + r do vat khdc nhau Co hao nhicu each phdn cdc do vdt
do thdnh 3 nhotn, sao cho nhom thit nhat chtia p do vat, nhom thti hai chua q
do vdt va nhom thii ba chiia r do vat
Loi giai
Socachphaiamla: P (p,q,r) = ^ ^ ^ ^ - 5 - ^
p!q!r!
Vi du 3: Co 8 nguai trong thang mdy cua mot ngdi nhd 6 tang Ho di ra theo
ba nhotn: 1 nguai, 3 nguai va 4 nguai Hoi c6 hao nhieu each thuc hien viec do,
neu a moi tang chi eo mot nhom di ra, va thii tu di ra cua cdc nguai trong mot
nhom khong c6 y nghia gi?
la: P8(1,3,4)C^ =280.20 = 5600 each
Vi du 4: Vol cdc chit soO, 1, 2, 3, 4, 5 eo the lap dugc bao nhieu so c6 8 chit
so, trong do chit sol lap lai 3 Idn, cdn cdc chit so khdc eo mat ddng 1 Idn?
Loi giai
So tat ca cac so c6 8 chu so trong do chu so 1 lap lai 3 Ian, con cac chu so
khac CO mat chi mot Ian la
QI p i
Pg (3,1,1,1,1,1)= =
-^ 3!1!1!1!1!1! 3!
Trong cac so tren day phai loai di cac so bat dau bang chu so 0, tuc la cac so
chi CO 7 chii so So cac so nhu the bang Py (3,1,1,1,1) = —
Vi du 5: Vai cdc chU soO, 1, 2, 3, 4, 5 eo the lap dugc bao nhieu sogom 9 chit
so', trong do mot chit so 0, 1, 2, 3 xudt him dung mot Idn, chit so"4 xuat hien
ddng 2 Idn va chU so 5 xuat hien ddng 3 Idn:
Loi giai
9' Neu tinh ca truong hop chi> so 0 dung dau thi c6 '-—- = 30240 (so)
^ 3!2!(1)!'^
54
Cty TNHH MTV DWH Khang Vi?t
Ta can phai loai ra cac truang hop c6 chu so'O dung dau
- Chon chii so' dau tien la chii so 0, con lai 8 vi tri
- So' cac so' thoa man de bai va c6 chii so' 0 6 dau la so' hoan vj lap cap 9 kieu (3; 2; 1; 1; 1) ciia 5 phan tu khac nhau, bang ^ = 3360 (so) '
3!2!(1)'' •/ V^y so cac so thoa yeu cau de bai la 30240 - 3360 = 26880 (so)
Vi du 6: Vai 6 chit so'O, 1, 2, 3, 4, 5 co the lap dugc hao nhieu so chia het cho
5 gom 11 chit so, trong do chit sol eo mat 4 Idn, chU so 2 c6 mat 3 Idn, chit so'3
CO mat 2 Idn, chU so 4 co mat 1 Idn vd tong solan xuat hi^n ciia chU so'O va chit
so 5 Id 1
Loi giai ^ A „ „ , , , ,
De so' can lap x = aia2a3a4a5a5a7a8agaiQa^i chia het cho 5 thi x phai tan
cung bang chii so 0 hoac chij s6'5
Vi tong so Ian xuat hien trong x cua 0 va 5 bang 1 nen neu x tan ciing bang
0, thi 5 khong co mat va nguoc lai neu x tan ciang bang 5, thi chir so' 0 khong xuat hien Boi vay 3; ( l < i < lO) chi co the la mot trong nhiing chu so 1, 2, 3, 4
Bai vay so'kha nang lap phan dau do dai 103^82 aga^Q cua so'x bang so'hoan
vi lap ciia 10 phan tu thuoc 4 loai chii so': 1, 2, 3, 4 voi 1 xuat hien 4 Ian, 2 xuat
hien 3 Ian, 3 xuat hien 2 Ian va 4 xuat hien 1 Ian, se bang P (1, 2, 3, 4) Ngoai ra
a^i CO the nhan gia trj 0 hoac 5 nen so can tim se la
2P(1,2,3,4)= 2 = 25200
^ ' ' ' ' l!2!3!4!
6.2 H O A N VI V O N G Q U A N H ( H O A N VI T R O N )
6.2.1 Vidu dan ddt Moi 6 nguoi khach ngoi xung quanh mot ban tron Lieu
CO bao nhieu each sap xe'p?
Neu ta moi mot nguoi nao do ngoi vao mot vi trf bat ky, thi so each sap xe'p
5 nguoi con lai vao 5 vi tri danh cho hp se la 5! = 120 * Vay CO ca thay 120 each sap xe'p 6 nguoi xung quanh mot ban tron
6.2.2 Cdng thiic So hoan vi vong quanh cua n phan tu khac nhau (Qn)
^^gc tinh bang eong thuc Qn = (n -1)!
Vi du l:Mot hoi nghi ban tron co 5 nude tham gia: nuae Anh eo 3 dai bieu,
^hdp CO 5 dai bieu, Diic co 2 dai him, Nhat eo 3 dai bieu vd My co 4 dai bieu
^oi CO bao nhieu cdch sap xe'p cho ngoi cho moi dai bieu sao cho 2 nguai ciing
1^6'c tich deu ngoi canh nhau?
Loi giai
E>au tien sap xe'p khu vyc cho dai bieu tung nuoc Ta moi phai doan nao do '^goi vao cho truoc Khi do 4 phai doan con lai co 4! each sap xe'p
55
Trang 29Boi duang hoc sink gioi Todn to hap - rai rac, Nguyen Van Thong
Do'i v a i m o i each sap xep cae phai doan lai c6:
3! Cach sap xep cac dai bieu trong npi bp phai doan A n h ; 5! each sap xep eac
dai bieu trong n p i bp phai doan Phap; 2! each sap xep cac dai bieu trong npi bp
phai doan Due; 3! each sap xep eac dai bieu trong n p i bp phai doan Nhat; va 4!
each sap xep cac dai bieu trong npi bp phai doan M y
Boi vay so' each sap xep cho ngoi cho tat ca cac dai bieu de n h i i n g nguoj
cung quo'c tich ngoi canh nhau se bang: 4!3!5!2!3!4! = 4976640
Vi du 2: Co bao nhieu each sap xep 5 nam: A i , A 2 , A 3 , A 4 , A 5 vd 3 nu:
Bi,B2,B^ vdomgt ban trdnneii:
a) Khong CO dieu kien gi? , ,1 fcl '
b) Nam khong ngoi canh nil B^? '
c) Nii khong ngoi canh nhau?
L o i giai a) M o i each sap xep bat k i la mot hoan v i vong quanh cua 8 phan t u , nen so
each sap xep bang so' hoan v i vong quanh eiia 8 phan t u , nen bang
Qg =71 = 5040
b) 5 nam 2 nve k h o n g ke c6 (7 -1)! = 6! each sap xep
M o t trong n h i i n g p h u o n g an sap xep 5 nam va 2 nit khong ke B^ n h u H i n h
1 K h i do Bj CO the xep vao giira A2,B2, giua B2,A4 , giua A4,B3, giiia A5,B3,
giira A3, A5 N h u vay c6 (7 - 2) = 5 each sap xep B^ Boi vay c6 6!5 = 720 each
sap xep
c) Truoc he't xep 5 nam ngoi x u n g quanh ban tron So each sap xep nay bang
so hoan v i v o n g quanh ciia nam phan t u , bang (5 - 1 ) ! = 4! = 24 ( H i n h 2)
Co 5 each sap xe'p nix B^, 4 each sap xep nix B2 va 3 each sap xep n u B3 (vao
giiia cac nam)
Vay so each sap xe'p can t i m la : 415.4.3 = 1440
do m o i phan t u , la m o t trong m phan t u da eho
T?i du l : C h o hai phan t u khac nhau a va b cae to hpp c6 lap chap 3 cua hai
phan t u da cho la: aaa, aab, abb, bbb ! • • <
Vi da 2;Cac to h p p c6 lap chap 2 eiia 3 phan t u khac nhau a, b va c la
aa, ab, ae, bb, be, cc
^inh li 7.12:So to hpp c6 lap chap n eiia m phan t u , k i hieu la bang
Chung minh: Ta danh so m phan t u da eho t u 1 tai m M o t to hpp c6 lap chap n eiia m phan t u da cho se dupe hoan toan xae dinh, ne'u v a i m o i so hieu
tu 1 toi m, ta biet so'cac phan t u eo so'hieu ay trong to hpp da cho
Ta thiet lap m o t t u o n g u n g giiia cae to h p p c6 lap chap n cua m phan tir v o i
cae day g o m cac chvt so 1 va 0 n h u sau:
Xet m o t to h p p eo lap chap n cua m phan t u , trong do c6 k^ phan t u c6 so hieu 1, k2 phan t u eo so hieu 2, km phan t u eo so hieu m , (tue la to hpp eo lap da eho eo kieu la (ki,k2, ,kn,) ( k j +k2 + + k ^ = n ) ) Ta eho u n g v o i no
k|,lan k2,lan kn,,lan ^
day sau day: 11 1011 10 011 1 Ne'u m o t phan t u ai do khong eo mat trong to hpp, tue la ki = 0, thi ta khong vie't n h o m c h u so 1 t u o n g u n g
(Thi d u : Ne'u ta xet cac to h p p eo lap chap 2 cua ba ehii a, b, c:
aa, ab, ac, bb, be, cc > iS > > s 1
voi cac kieu theo t h u t u la: (2, 0, 0); (1,1, 0); (1, 0,1); (0, 2, 0); (0,1,1); (0, 0, 2) Thi cae day u n g v a i chung la: 1100,1010,1001, 0110, 0101, 0011)
So c h u so' 1 t h a m gia vao day u n g v o i to h p p kieu [k-^M2>->^m) ^^"8
k i + k2 + + k^n = n , eon so ehir so 0 thi bang m - 1
Ta nhan thay rang m o i day noi tren la m o t ehinh h p p eo lap ciia n chit so 1
v a m - l c h u soO
Dao lai, u n g m o i ehinh hpp c6 lap cua n chu so 1 va m - 1 c h u so 0; eo m o t
to hpp CO lap eiia m phan t u chap n ma ta c6 the vie't ra m p t each de dang
(Thi d u : Lfng v a i ehinh hpp c6 lap 011100111111 a i a 9 chir so 1 va 3 chu so 0' ta CO the lap lai to hpp c6 lap cua 4 phan t u a, b, e, d chap 9 n h u sau bbbdddddd)
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Trang 30Boi duang hQc sinh gioi Todn to hap - rai rac, Nguyen Van Thong
Nhu vay, so to hgp c6 lap chap n cua m phan tu bang so chinh hgp c6 lap
cua n chii so' 1 va m - 1 chu so 0 Ki hieu so to hop c6 lap chap n cua m phan t u
^ = ct5-i= — = 70
Vi du 3:Tit edc ehU sol, 2, 3, 4, 5 c6 the lap ducfc bao nhieu so, sao cho trong
moi so'do so'chit so< 5 vd edc chit soduoc sap xep theo thii tu khonggiant?
Lai giai Bai toan chung quy la tim tong so cac to hgp c6 lap chap 1, chap 2, chap 3,
chap 4, chap 5 cua 5 chii so da cho
: Vi: Cl=Cl,_,=Cl=5,cl = Cl,_,=Cl=15;
^5 = C3+5-1 = C7 = 35X5 = Cl^s-i = Cg = 70,
^5 = C 5 + 5_ i =Cg =126
, nen c6 ca thay 5 +15 + 35 + 70 +126 = 251 so thoa man cac dieu ki$n cua dau bai
^di todn chia khoa 5: Co bao nhieu each phan phoi n do vat giong nhau cho
p nguoi?
Loi giai
Gia su nguoi thu nha't nhan dugc do vat, nguoi thu hai nhan dugc kj
do vat, nguoi thu p nhan dugc kp do vat (k^ + k 2 + + kp = n)(ki >0) Ta
vie't tugng trung each phan phoi do nhu sau:
l L J 2 2 ^ p p p k|l5n k2lan ^ ^ ^ ^ (Ne'u kj = 0(i = l, ,p) thi ta khong vie't day chir so i tuong ung)
Nhu vay moi each phan phoi ung voi mot to hgp eo lap chap n eiia p phan
hi', dao lai, ung voi moi to hgp c6 lap chap n ciia p phan tu, c6 mgt each phan
pho'i n do vat gio'ng nhau cho p nguoi
Cty TNHH MTV DWH Khang Vi?t
Vay so'each phan phoi bang so to hgp c6 lap chap n ciia p phan tu, tue la
^di todn chia khoa 6;Phuong trinh + X2 + + x^, = n , trong do n la mgt so
nhien, eo bao nhieu nghiem tu nhien?
Loi giai Ne'u ( k i , k 2 , , k m ) la mgt nghiem ty nhien ciia phuong trinh da cho, thi ta
CO the cho ling voi no mgt to hgp eo lap chap n ciia m phan tu
k i e u ( k i , k 2 , v k m ) Dao lai, ne'u c6 mgt to hgp eo lap ciia m phan tu chap n kieu (ki,k2, ,k^)
thi ta tim dugc mgt nghiem tu nhien ciia phuong trinh da cho bang each dat
X i = k i , X 2 =k2, Xm = k n ,
, Vay so nghiem tu nhien ciia phuong trinh da cho la so to hgp c6 lap chap n
ciiam phan tii, hie la Cj^ = Cl^^_-^ = ^".^"^ ^}'
n(^m-lj!
NGUYfiN BU T R 0 TONG Q U A T
Midnxet
Mgt lop toan rai rac gom 30 nix sinh va 50 sinh vien nam thu hai Hoi eo bao
nhieu sinh vien trong lop la nii sinh hoac la sinh vien nam thii hai? Cau hoi nay khong the tra lai dugc trir khi cho them mgt so thong tin nira Cong so nir sinh voi so' sinh vien nam thu hai c6 the se khong eho cau tra loi diing boi vi so' sinh vien n i i nam thii hai se dugc tinh hai Ian Chinh vi vay so'sinh vien trong
lop hoac la nix hoac la sinh vien nam thii hai la tong so sinh vien nu va so sinh
vien nam thu hai tru di so sinh vien nir nam thii hai Ki thuat giai bai toan de'm nhu the da dugc gioi thi^u Trong mue nay ta se tong quat hoa nhiing y tuong
<Ja dugc dua vao trong mue do de giai mgt lop rgng hon niia cac bai toan de'm
J^guyen ly bii trit
Co bao nhieu phan t i i trong hgp ciia hai tap hgp hiiu han phan tii? Ta da chung to rang so cae phan t i i trong hai tap hgp A va B bang tong cac phan tu
Ciia moi tap trir di so phan tii eiia giao hai tap hgp, tiic la:
AuB| = |A| + |B|-|AnB Nhu da chi ra, eong thiie cho so cac phan tu eiia hgp hai tap hgp rat hay dung trong cac bai toan de'm
Trong phan sau ciia muc nay, ehiing ta dua ra each tim so cae phan t i i ciia hgp mgt so hiiu han cac tap hgp ket qua tim dugc khi do dugc ggi la nguyen
59
Trang 31Boi dumtg hoc sitth gioi Todn to hop - rbi nic, Nguyen Van Thong
ly bu tru Truac khi nghien cuu hgp ciia n tap hgp trong do n la mgt so'nguyen
duong tuy y, chung ta se trinh bay each rut ra cong thiic tinh so' phan t u ciia
hgp ba tap hgp A, B, C Truoc khi xay dung cong thuc nay ta luu y rang
A| + B| + C dem mgt Ian nhiing phan tu chi thugc mgt trong ba tap, dem hai
Ian nhiing phan t u thugc dung hai trong ba tap va dem ba Ian nhung phan tu
thugc ca ba tap.(xem minh hga tren hinh 4(a))
De loai bo viec dem trung lap cac phan tu thugc nhieu han mgt tap, caVi
phai tru di so' cac phan t u thugc giao cua tat ca cac cap cua ba tap hgp nay, tuc
lanhandugc: |A| + |B| + |C|-|AnB|-|AnC|-|BnC| "' ' * -'^^ '
Bieu thuc nay dem mgt Ian cac phan tu chi thugc dung mgt trong ba tap da
cho Cac phan t u xua't hien trong dung hai tap ciing dugc dem mgt Ian Tuy
nhien, bieu thuc tren chua dem cac phan tu thugc ca ba tap Ian nao, vi chiing
CO mat trong ca ba giao ciia cac tap dieu nay dugc minh hga tren hinh 4(b)
De khoi bo sot, ta them vao so cac phan tu thugc giao cua ba tap bieu thiic
cuo'i cung nay tinh moi phan t u diing mgt Ian du no thugc mgt hay hai hoac ca
ba tap Do do: A u B u C = A + B|+ C -|AnB - A n C - BnC + A n B n C
Cong thuc nay dugc minh hga tren hinh 4(c)
(a) Dem cac phan t u theo
6n
CtyTNHHMI i yviIKhang Viet
ft du 1 Biet rang c6 1232 sink vim hoc tieng Tdy Ban Nha, 879 sink vim
c tieng Phdp vd 114 sink vim hoc tieng Nga Ngodi ra cdn hiei rang 103 sinh Jiett hQC cd tieng Tdy Ban Nha vd tieng Phdp, 23 sinh vim hoc ca tieng Tdy pan '^^ "^^ ^^"^ ^^^^
2092 sinh vim dm theo hoc it nhat mot ngoai ngU, thi c6 bao nhieu sinh vim
hoc cabathii tieng?
Ggi S la tap sinh vien hgc tieng Tay Ban Nha, F la tap cac sinh vien hoc tieng Phap, R la tap cac sinh vien hgc tieng Nga Khi do:
|S| = 1232,|F| = 879,|R| = 114,|SnF| = 103,|SnR| = 23,|FnR| = 14
va |SuFuR| = 2092 Thay tat ca cac dai lugng nay vao cong thuc tong quat:
SuFuR| = |S| + |F| + |R|-|SnF|-|SnR|-|FnR| + |SnFnR| tanhandugc
2092 = 1232 + 879 + 114-103 - 2 3 - 1 4 + |SnFnR Giai ra ta dugc |S n F n R| = 7
Do vay, c6 7 sinh vien theo hgc ca ba thu tieng Xem minh hga tren hinh 5
Hinh 5 Tap cac sinh vien hgc tieng Tay Ban Nha, Tieng Phap va Tieng Nga
^inh ly I.Nguyen ly bu tru Cho Ai,A2, ,A„ la cac tap huu han Khi do
A i u A 2 U u A „ | = X - S
lsi<n l<ii<i2^n
\2
I<il<i2<i3^n n+1
Trang 32Boi duang hpc sitth gioi Todtt tohgrp - riri r^c, Nguyen VSn Thdttg
dem C (r, 1), Ian trong ^ | A i | N o dugc dein C (r, 2) Ian trong ^ J A j n Aj| Tonj^
quat, no dugc dem C (r, m ) Ian boi tong c6 lien quan toi m tap A j Vay phan tu
nay dugc dem d u n g bang
C ( r , l ) - C ( r , 2 ) + C(r,3]- + (-l)'"^^C(r,r) Ian boi v e p h a i cua cong thuc trong
d i n h ly M y c dich cua ta la tinh gia t r i ciia dai luong nay Ta c6:
C(r,0) - C(r,l) + C{r,2) - C(r,3)+ + (-!)'• C(r,r) = 0
Suy ra 1 -C(r,0) = CCr.l) - C(r,2) + C(r,3) - + (-ir ^C(r,r)
D o do, m o i phan t u cua hop dugc dem dung m p t Ian k h i tinh gia t r i 6 ve
phai cua cong thuc da cho Vay nguyen ly bu t r u da dug-c chiing m i n h
N g u y e n ly b u trur cho ta cong thuc tinh so phan t u cua hgp n tap hg^p voi
m p i n nguyen d u o n g Trong cong thuc do c6 cac so hang cho so phan t u trong
giao cua m p i tap con khong rong thuQC tap gom n tap hgp Do do, cong thuc
nay c6 2" - 1 so'hang
Cdch2:
Ta chung minh cong thuc (1) b ^ g phuong phap quy nap theo so n cac tap hgp
n = 1, 2, cong thuc (1) da diing
Gia su cong thuc (1) d i i n g den n tap hgp h i m han, khac rong cho truoc
Xet n + 1 tap hgp h i h i han bat ky: Ai,A2, ,A„+i thi ta c6:
Cty TNHH MTV DVVU Khaiig Vict
n+1 ( n
- Theo t r u o n g h g p n = 2:
- Theo gia thiet q u y nap:
n+l i=l i=l U A i IjAi
T?i du 2: Co 15 qua cau doi khac nhau, trong do c6 4 qua mau vang, 5 qua
0au xanh, 6 qua mau do Co bao nhieu each chon ra 10 qua sao cho trong cac qua cau con lai c6 du ba mau?
L o i giai:
Gpi A la tap hgp cac each chgn ra 10 qua cau; V, X, D theo t h u t u la tap hgp
cac each chgn ra 10 qua cau ma trong cac qua cau con lai k h o n g c6 qua nao mau vang, xanh, do; n la so each chgn 10 qua cau thoa m a n yeu cau Ta c6:
Vay: n =cj^ -(cfi -Cfo -C^ - 0 - l + o) = 3003 - (462 + 252 + 126 - 7) = 2170
Vidu 3: Cho 3 chUso: 1,1, 3 Txtcdc chUsotren c6 the lap dU(?c bao nhieu so
gom n chic so (n S 3) sao cho moi so do, c6 mat cd ba chit sol, 2, 3
Loi giai Ggi X la tap h g p cac so' thoa man dieu kien bai toan T la tap h g p cac so' gom
n chu so' duge lap tir ba chir so' da cho Ti la tap h g p cac so' thugc T ma trong so'
do khong CO mat chir so i (i = 1, 2, 3)
T a c o : |x| = |T|-|TiUT2uT3
T| = 3"
T1UT2UT3 Tj nT2 - T2 nTg - T3 n T 4 + T^ nT2 nT3
De tinh dugc: \\1 = |T2 nT31 = IT3 nT4J = 1;|TI n T 2 n T31 = 0
V?iy: |TI u T2 u T3I = 3.2" - 3 => |X| - 3" - 3.2" + 3
^ ' d y 4:Cho n Id so nguyen duong vd cho k so nguyen duang aj ,a2, ,a|^ doi
^Qt nguyen tocung nhau Tim so cac so nguyen duomg a < n sao cho a khong '^hia hei cho a, vai mgi i = 1,2, , k
•••itasi L a i giai ; ' •
^ o i i = 1, 2, , k Ta dat Aj = |a e N* /a ^ n,a iaj I va A la tap hgp cac so thoa
(5) "^an dieu kien cua bai toan t h i |A = n -U A i k
t>ethay A j = { a i , 2 a i mjaj) v o i i t i j e N * va thoa m a n : mjaj < n < ( m i + l ) a i
63
Trang 33Boi dumtg hoc sink gioi Todn to hgp - roi rac, Nguyen Van Thong
l<il<i2<kLaiiai2
k
L i = l J
Vi du 5: Cho m qua cdu doi mot khdc nhau vd n cdi hgp dot mgt khdc nhau
(m > n) Co bao nhieu each bo tat cd cdc qud cdu vdo trong cdc hop sao cho hop
ndo cung c6 it nhat 1 qud cdu (khong kethii tu cdc qud cdu trong moi hop)
Ldigiai Khi giai bai toan nay, rat nhieu hoc sinh mac sai lam nhu sau: Truoc het i
vao moi hop mpt qua cau: Co each, sau do phan phoi ngau nhien m
-cau con lai vao n cai hop va c6 n " " " each va ke't luan: AjJ^.n"""" each! Ca
giai tren da mac sai lam vi ta khong kethif tu ciia cdc qud cau trong moi hop
De giai bai toan nay, ta phai diing phuong phap "gop vao va loai di"
Gpi A la so each bo cau vao hop thoa man dieu kien bai toan;
Vai moi i = 1, 2, , n k i hieu Aj la so' each bo cau vao hpp ma A; kho;
chiia cau S la tap hgp tat ca cac each bo cau vao hpp the thi A = S I j A i
i=l
Ta de thay so' each bo ngau nhien x qua cau vao y cai hpp (so' lugng cau tror
moi hop khong han che va khong ke thu tu eae qua cau trong moi hgp) la y"
That vay: Qua cau thii nhat c6 y each bo vao hop (bo vao hpp cung duo;
qua cau thii hai cung c6 y each bo vao hop (bo vao hop nao eiing duoc, ke •
hop da CO cau) ; qua cau thii x eiing eo y each bo vao
Vay theo quy tac nhan thi ta c6: y.y y = y^ each bo het cau vao hpp
Ail
Cty TNHH MTV DV \ hliang Vi$t
Xir nhan xet tren ta thay: |S| = n™, |Ai| = (n -1)"" (bo m qua cau vao n - 1 cai j^^p khac hop A;) voi i = 1, 2, , n
js^^nA^ ={n-2y" (bo m qua cau vao n - 2 cai hop khac hop Aj va Aj),
J^Ihdnxet:Bdn chat bdi todn tren Id "Co bao nhieu toan dnhf: X -> Y, trong do X
gom com phan tie con Y CO n phan tu, m>n"
^ditodnchiakhda?:
Cho hai tap hap him han khdc rong c6 cung sophdn tvt:
X={ai,a2, an} vd Y=:(bj,b2, b„} Hay tim so cdc song dnh f: X -> Ysao
cho f(aj);tbi,Vi = l,2, ,n
Lai giai
Ta ky hieu A la tap cac song anh f : X - > Y thoa man dieu kien bai toan; Voi moi i = 1, 2, , n Ta dat A i la tap hpp cac song anh f : X Y ma f(ai) = bi; S la tip hpp cac song anh t u X len Y t; i ' • ^
Ta CO nhan xet: Neu eo 2 tap hpp khac rong, huu han P, Q eo eiing so'phan tir la k thi CO k! song anh tu P len Q (moi song anh tuang ung voi 1 hoan vj ciia
k phan tu) T u nhan x ?t do de thay:
Ai| = (n-1)!, V i ' 1,2, ,n :Ai^nAi2| = ( n - 2 ) ! , V i ^ j , i,j = 1, 2, n
n A j 2 n n A i | J - ( n - k ) ! , Vii,i2 ik thoa 1 < i j < 12 < < ^ n Vay:|A|=:n!-Ci(n-l)!+c2(n-2)!- + ( - l ) " c r H n - n ) !
1! 2! ^ ^ " n!
1 1 (-1)"
+ + -——
Trang 34Boi duinig hoc siiih gidi Todii idhgtp - rai rac, Nguyen Van Thong Cty TNHH MTV DWH Khang Vi$t
Theo (1.1) ta c6:
A i U A i
Vi du 6:MQt nguai dua thu phdn phot n bite thu tnattg ten ngucn nhdn khdc
nhau vdo n hop thu cua cdc ngu&i nhdn Do ten nguai nhdn ghi tren cdc phong
bi bi ma nen viec phdn phot duoc thycc hien ngdu nhien moi thu vdo mot hop,
Co bao nhieu kit qua phdn phot khdc nhau, sao cho c6 it nhdt mot btic thu den
dung dia chi?
Ta CO the phat bieu bai toan 6 dang khac n h u sau: ,
<Bdi todn chia khoa 8:
Cho hai tap hap him hqn khdc rong c6 ciing so phan tic : \ [a-^,^2,->^n\"
Y = {bi,b2, ,bn} Hay tttn so cdc song dnh f: X Y sao cho ton tai i s (l,2, ,n} ma
chung cua k h o n g it h a n m trong so cac tap Ai,A2, ,A„ v o i
(**)
k
<^di todn chia khoa 9:
Cho hai tap h g p hiru han, khac rong c6 ciing so phan t u la n: X={ai,a2, ,a„}
Va Y = {bi,b2, ,bn} Co bao nhieu song anh f : X ->• Y sao cho c6 d u n g k so
t u n h i e n i i , i 2 , i k ( l 2 k < n ) ma f aj = b j , Vj = 1, 2, k
Loi giai
Gia sir ngugc lai, m o i phan tir ciia A^ deu thuoc khong qua m trong so'cac
tap Ai,A2, ,An v a i m<^^-—^ Tir (*) suy ra m o i ai thuoc A^ (1 < i < k) t u o n g
K
ling v o i k h o n g qua m m <-n - 1 tap khac nhau trong so'cac tap A2,A3, ,A„
n - 1 Boi vay so'cac tap da cho se nho h a n k + 1 = n
v o i cac bj.)
Ta CO ( n - k ) !
n-k
Ne'u m < n - 1 , t h i se ton tai Aj (1 < j < n) ma a k h o n g thupc A j K y hieu phan
Truoc he't ta chon k so trong tap X: a;^ ,a,^ a,^ va dat {{a,.) - b j , j = 1, 2, tir chung d u y nhat cua Aj v a i A^ bang b v o i Ajt (1 < t < m ) bang a^^ K h i do
k coCn each chon k phan tir n h u vay '''t (1 ^ t < m) (ajt b) va Vt, s (1 < S, t < m) (s ^ t =>ais ; t a i t )
Ta xet n - k phan tir con lai trong X va n - k phan tir con lai trong Y (khac That vay, ne'u 3t (1 < t < m) ma aj^ = b , t h i A j va A i t c6 it nhat hai phan tir
chung la a va b, nen m a u thuan v a i (*) Ne'u 3s.t (1 < s.t < m ) ma ajj = a n h u n g
^is = , t h i Ajj va A^j c6 it nhat hai phan tir chung la a va a^^, nen m a u thuan
song anh tir X len Y ma f(aj.) = b j Voi
Do do Aj chua it nhat m + 1 phan tir khac nhau: b,aii,ai2,-,ain, Aj > m + l > k
^ aa d i toi m a u thuan v o i gia thiet a) Bai vay, m = n - l v a P | A j = { a } '
(Theo quy tac nhan)
Vi du 7:Cho cdc nguyen duong k, n thoa man dieu kien n>k^-k + 1 Gia sii \ i (1 < i < n) d u n g ai, (1 < t < - 1) de ky hicu car ph^n t-V khac a % a
fl) |Aj| = k W ( l < j < « ) Aj={a,ai,a^ a,U)
2! 3! ( n - k ) !
Vay so song anh can t i m la: CJ^(n-k)! i_i C-i)"'*'
2! 3!"^""^ ( n - k ) !
Trang 35B^i duang hifc sinh gidi Todn td'htpp - riri r^c, Nguyen Van Thong
Khi do Vi, j (1 < i, j < n) neu i ?t j thi Vt (1 < t < k - 1) a[ That v^iy, gia si,
3i,j (1 < i , j < n) ma i ^ j, nhimg 3t (1 < t < k - 1) (a[ 5^ a't), thi A i , Aj c6 it nhat
hai phan t u chung la a va Ta da di den mau thuan voi (*)
N h u vay Vi.j (1 < i, j < n), ma i ^ j thi Ai, Aj chi c6 mpt phan t u chung duy
<Bdi 1 (Ti§p Khac) Hay tim so'cac cap khdc nhau cua cac tap con khon^
giao nhau thugc tap gom n phdn tii : =
Loi giai
Gia su M la tap gom n phan t u va A, B la hai tap con tuy y khong giao nhau
cua tap M Trong cap a = (A, B) ta quy dinh tap ben trai (A) la tap thii nhat, tap
ben phai (B) la tap thu hai Ta tim tat ca cac cap dugc tinh thii t y cua cac tap
con thupc M khong giao nhau • , s
Xet phan t u a tiiy y thuQC tap M Co ba kha nang xay ra: hoac a thupc tap
thu nhat (a e A), hoac a thupc tap thu hai (a e B), hoac a khong thupc tap thii
nhat Cling nhu tap thu hai (a ^ A, a i B) Nhu vay so cap dupe sap thii t y gom
CO cac tap con khong giao nhau ciia tap gom n phan t u se la 3" Trong so' cac
cap nay c6 mot cap duy nhat, ma hai tap con thupc no deu rong nen c6 3" - 1
cap ma moi cap nay c6 it nhat mpt tap con khac rong
Doi vod moi cap a = (A, B) trong 3" - 1 cap ke tren ta doi cho tap con thu nhat
va thu hai cho nhau se dupe cap a' = (B, A), nhung ve thuc chat a, a' ciang chi 1^
mpt cap thoa man dieu ki^n AnB^^tQ Boi v^y ta chi c6 (3" - 1 ) : 2 cap kh
nhau gom hai tap con khong giao nhau, trong do c6 it nhat mpt tap con khJ"^^
rong Ngoai ra cap gom hai tap rong cung thoa man dieu kien khong giao nhau
3 " + l Boi vay so' cap thoa man yeu cau ciia dau bai se la: (3" - 1 ) : 2 +1 = —^—
^ d i 2(Ba Lan) Cho tap Mgom n phdn tt( Voi hai tap con tiiy y A,B ciia tap A'
tinh so'phdn tic cua giao A n B Chiatg minh rang tong cua tat ca cac phdn i"
cua mgi giao c6 the gom hai tap con cua tap M bang n.4"~^ ?
Vi moi phan tu ciia tap M chi thupc hoac tap con A hoac tap con A Boi vay
no chi thupc mpt trong bo'n tap Ciia bp 4:
A n B , A n B , A n B , A n B (**)
Bai vay so phan tu ciia cac tap thupc mpt bp 4 tiiy y (**) se la n
Do CO, 4""^ bp 4 (*) khac nhau, nen c6 4"'^ bp 4 (**)
Vay tong so phan t u ciia tat ca cac tap dang A n B la n.4"~^ '
<6di 3 (My) Voi so tu nhien tiiy y n > 3 cho k = - n ( n + l )
o va tap Xn gom
n(n + l)/2 phdn ti, trong do c6 k phdn tit mau xanh, k phdn tit mau do, cac
phdn tit con lai deu mau trang
Chung minh rang: Co the chia tap X thanh n tap con titng cap khong giao
"hau Ai,A2, ,An sao cho voi m so'tuy y (l<m<n) cac tap con A,,, gom ddng
w phdn tii va cac phdn tit deu citng mau
' ' • Lai giai Bai toan dupe chung minh bang quy nap theo n > 3 ' r > ;
1) Ca sa quy nap. Voi n = 4, 5, 6, 7, 8, 9 chia XjCl < i <0) dupe chi dan trong ''ang sau:
So phan
tu mau
do (Cac nhom mau do)
So phan tu mau trang (Cac nhom mau trang)
6 9
Trang 36Boi duaitg hqc sink gidi Toatt tohgrp - red rac, Nguyen Van Thong
2J Qu(/ nap Gia su n > 10 va doi v o i cac so nho h o n n, cu the la doi v o i n - 6
khang d i n h da dugc chiing m i n h K h i do theo gia thiet quy nap, tap con
( n - 6 ) ( n - 5 3
„ V ( n - 6 ) ( n - 5 ) , , ^, , ,
mau xanh, = n ( n - 6 ) ( n - 5 ] phan t u mau do, con lai la mau trang da dugc
chia thanh n - 6 tap con: A^, A 2 A ^ g ma moi tap con deu gom cac phan h i cung
mau va luc lugng cua moi tap con deu bang chi so hrang ung cua tap con nay
[ n - 6 ) ( n - 5 ] 1 ^ n [ n + l ) ( n - 6 ) C n - 5 )
6 6 6 "
Nen k - k i = 2n - 5 T u do suy ra so l u g n g phan h i cua Xn mau xanh va mau
do nam ngoai X„_g deu bang 2n - 5
O n Y Y - " C n - 1 ) ( n - 6 ) ( n - 5 ) _ ^, ,
u o A n - An_6 = 6 n - 1 5 , nen so l u g n g phan t u mau
trang cua Xn nam ngoai X „ _ 6 la 6n - 1 5 - 2(2n - 5) = 2n - 5
K h i do xac d j n h cac tap con A n _ 5 , A n _ 4 , A n _ 3 , A „ _ 2 , A n _ i , A n n h u s a u :
^ n - S ' ^ n g o m cac phan t u mau xanh,
A„_3,Aj,_2 g o m cac phan t u mau do,
; A n - 4 ' A n - i g o m cac phan t u mau trSng
70
Cty TNHH MTV DWH Khang Vi?t
4 (Nam Tu-) Dot vai moi sotu nhien n e N hay tim sotu nhien k e N Ion
fihat, thoa man dieu kien: Trong tap gom n phan tit c6 the chon ra dugc k tap
t con khdc nhau, md hai tap hat ky trong cac tap con nay deu c6 giao khdc rong
Lai giai
J G i a s u X = {ai,a2, a„} •, >
1) Chung minh k>2"~^ Ta co djnh phan t u va chi xet cac tap con
A i , A 2 - A k CO chua K h i do so tap con nay bang d u n g so tap con cua tap
Do do giao cua hai tap nay bang rong Vay ta di toi mau thuan v o i tinh chat
cua cac cap da chgn ra, nen k<2"~^.Vay: k = 2"''^ ^ 'j-jy,'!
to5(Anh) Gia sutrongtap hOu hanXchgn dugc50 tap con AI.A2,.-,A^Q, md
moi tap con nay deu chiia qua nua so'phdn tuciia tap X
Chihig minh rang co the tim dugc tap con B c X chua khong qua 5 phan tu
vd CO it nhat mot phan tit chung voi titng tap Aj (l<i< 50)
Loi giai 1) T u dac diem ciia cac tap Aj (1 < i < 50) da chgn ra co khang d j n h : V o i m g i
so'nguyen k < 50 trong cac k tap con da chgn ra A^i.A^j.-Aiii luon l u o n tach
f k l dugc khong it h o n — + 1 tap co phan t u chung
That vay, neu chi co khong qua
2 CO phan tir chung, thi m o i phan tir thugc cac tap A j ( ( l < t < k) dugc tinh khong qua Ian K h i do tong so phan tu' ciia ta't ca cac tap con Ajj(1 < t < k) se khong v u g t qua
Trang 37Boi duSng hoc sink gioi Todn toh<;rp - riri rac, Nguyen Van Thong
So sanh (1) va (2) ta d i toi mau thuan, nen trong k tap con tren luon tach ra
f k l
dugc — + 1 tap c6 phan t u chung Khang d i n h dugc chung m i n h ;
_ 2 J • J > : if
2) Chpn cac phan t u ciia tap B * ' ~
Xuat phat t u 50 tap da chgn ra K h i do khang d i n h tren u n g v d i k = 50 ta
tach dugc 26 tap con c6 phan t u chung K i hi^u phan t u chung nay bang
Loai cac tap chua , con lai 24 tap k i hieu cac tap nay la: Aji,Aj2, Aj24
K h i do, theo khang d j n h tren, u n g v o i k = 24 ta lai tach dugc 13 tap c6 phan
t u chung K i hieu phan t u nay la 82 Loai bo 13 tap A j t ( l < t < 24) c6 chua 82
con 11 tap K i hieu cac tap nay la: Api,Ap2, ,Apii. - > • > • « » : • • » - W V A *
K h i do, theo khSng d j n h tren ta lai tach dugc tap con c6 chua phan tir
chung K i hi^u phan t u nay bang 8 3 Loai bo cac tap A p t ( l < t < 11) c6 chua 83
con 5 tap K i hi^u cac tap nay la: Aqi,Aq2,Aq3.Ac,4,Aq5
K h i do, theo khang d j n h tren, u n g v a i k = 5 ta lai tach dugc 3 tap c6 phan t u
chung K i hi^u phan t u nay bang 8 4 Loai bo 3 tap chua 84 con 2 tap a
VI m o i tap nay deu chua nhieu h o n ^ phan t u , nen chung phai c6 phan t u
chung K i hieu phan t u nay la 8 5
Tap B = { 8 1 , 8 2 , 8 3 , 8 4 , 8 5 } CO phan t u chung v d i t u n g tap A i ( ; i < i < 5 0 ) da
chgn ra
Cac phan t u a^ , 8 2 , 8 3 , 8 4 , 8 5 c6 the triing nhau nen B chua khong qua 5 phan tu
5gdt 6 (Ao - Ba Lan) Cho 1978 tap hop ma moi tap nay dm chtca dung 40 phdn
tic Biei rang hai tap tuy y trong cac tap nay deu c6 dung mot phdn tu chung
Hay chUng minh rang ton tai phdn tit thuoc toi cd 1978 tap da cho
L a i giai fi
'''^^ Xet tap A tuy y trong 1978 tap da cho.' l>^> } ''-llv n fifef• *•>••10
V i A CO phan t u chung v o i t u n g tap trong 1977 tap con lai, nen trong A phai
ton tai phan t u a nao no thugc it nhat 50 trong 1977 tap con lai (Ngugc lai, neu
mot trong 40 phan t u cua A chi thugc 49 tap con lai, t h i so tap da cho khac A se
la: 4 0 x 4 9 = 1960< 1977)
N h u v a y , p h a n t u a t h u g c 51 tap A,Ai,A2, ,A5o r ^ ffit^
C h u n g m i n h a thuOc tap bat k y B trong 1978 tap da cho
V i hai tap tuy y trong cac tap da cho c6 d i i n g m o t phan t u chung, nen cac
tap A,Ai,A2, ,A5o k h o n g the CO phan t u chung nao khac a
Cty TNHH MTV DWH Khang Viet
Gia su phan t u a khong thugc B K h i do v a i mSi A i ( l < i < 5 0 ) B phai c6
phan t u chung a^^h va cac phan t u nay cijng phai khac nhau (Neu 8; = Sj thi
^., A j ( l ^ i, j ^ 50) se CO it nhat hai phan t u chung la a va 8 ; ) Boi vay tap B chua Idiong it h a n 51 phan t u Ta da d i toi mau thuan v a i dieu kien D o do a thugc t$p B
V i B la tap bat k i trong 1927 t^p con l ^ i B ?t A,B ?t AjCl < i < 50), nen a thugc
tat ca 1978 tap da cho. t^^^ :,y
^di 7 ( T r u n g Quoc) Tap hop gom 2" phdn td dugc chia thanh cac tap con doi
ftigt khong giao nhau Xet phep hien doi: Chuyen mot so'phdn td tit tap con ndu sang tap con khac, trong do mot so phdn tit dugc chuyen cdn phai hang so phdn
tit hi?n CO cua tap nhan (di nhien phep hien doi chi dugc thuc hien doi vai nhvmg
cap ntd luc lugng cua tap chuyen khong nho ham luc lugng cua tap nhdn)
Chiing minh rang hang mgt sohihi han huac thuc hien phep hien doi tren se dugc tap con triing vdi tap xuat phat ' * * ' *? ; •
„ •,, , L a i giai • -s v '
Xet cac tap con c6 luc l u g n g la so le (neu c6) ;is m^n fr^v
V i luc l u g n g cua tap xuat phat chan, nen so tap con c6 luc l u g n g le phai la
so chan Ta chia cac tap c6 luc l u g n g le mgt each tiiy y thanh cac cap
D o i v d i m o i cap g o m 2 tap con cd luc l u g n g le ma ta thuc hien phep bien doi dua ra trong bai toan bang each chuyen so phan t u bang luc l u g n g tap nho tir tap Idn sang tap nho Sau k h i hoan thanh phep bien d o i nay d o i v d i tat ca cac cap tap con cd lye l u g n g le, ta dugc cac tap con deu cd luc l u g n g c h i n
V d i n = 1 tap xuat g o m 2 phan t u , nen ca 2 phan t u nay deu nam trong mgt tap con
Vdi n > 2 xet tat ca cac tap con v d i luc l u g n g khong chia het cho 4 V i luc
l u g n g cua tap xuat phat chia het cho 4, nen so cac tap cd luc l u g n g khong chia het cho 4 phai la so chan
Ta chia m g t each tiiy y cac tap nay thanh cac cap D o i v d i m o i cap nay lai thuc hien phep bien doi de xuat trong bai toan Cudi cung dugc tat ca cac tap con deu cd luc l u g n g chia het cho 4
Doi v d i mgt cap tap con cd luc l u g n g chia het cho 4 ma chuyen so phan t u bang luc l u g n g tap con tir tap Idn sang tap nho, t h i tap nho se thanh tap con co luc l u g n g chia het cho 8
Thuc hi?n bien doi t u o n g t u cuoi ciing dugc mgt day tap con m a luc l u g n g cua chung chia het cho 8,16, 32, tuc la luc l u g n g cua chung chia het cho 2 ,
t = 3, 4, 5, n, Bdi vay toan bg 2" phan t u ciia tap xuat phat phai nam trong mgt tap con
73
Trang 38Boi duong hoc sink gioi Toan to hap - riri rac, Nguyen Van Thong
<6di 8 (Tiep Khac) Gid sii M Id tap con cua tap gom tat cd cdc cap sotu nhien i ^
k khong vuat qud sotu nhien n(n>2) cho truac Trong do, neu cdpi<k thuoc tap
M, thi khong mgt cdpk<m thuoc M Hay xdc dinh solan nhai cdc cap cd the CQ
trong tap M?
••:S i^:::K^i'-il'iHii}.l"':"i"/9f^!i-'\ „ T ,.„; • , i f>6 t:,i
Loi giai Dung A, B de ki hieu mot each tuong ung, tap gom tat ca cac so' nho va tap
gom tat ca cac so Ion, ma chung lap thanh cac cap trong M Khi do, theo dieu
ki^n bai toan, khong mot phan tu nao ciia B lai thuoc A, nghla la A n B = 0
Gia su luc lugng cua cac tap A, B bang a, b
Khi do do AuBc{l,2,3,4, ,n-l,n} nen a + b < n va cap bat ky thuoc M
phan til" nho (so nho) nhan khong qua a gia tn, con phan t u Ian (so Ion) nhan
khong qua b gia trj Boi vay so phan tu ciia t^p M khon^ vup't q^ua
truonghop: M = |o;l)/j<|;! (khi do M gom cap)
Nell so n le, thi can chon M = |o;l)\j<|;l > | | (khi do M gom
n - 1 n + 1
4 4 phan tu)
N h u vay, voi n chan hay le so cap Ian nha't c6 the c6 trong tap M se la
•
<Bdi 9 (IMO Ian thu32 nam 1991)
Cho S = {1;2; ;280} Tim sotu nhien nho nhat n sao cho moi tap hap con n
phan tit cua S deu chua 5 sotu nhien nguyen tociing nhau titng doi mot
Loi giai s ''
^' Trudc he't ta dua vao nguyen ly chua - chua trong de chung minh rang n >
217 Gpi Ai,A2,A3,A4 la cac tap hgp con ciia S chiia cac bpi Ian lupt ciia 2, 3, 5,
7 Liicdo |Ai| = 140, |A2| = 93, |A3| = 56, |A4| = 40, | A i n A 2 | = 46, | A i n A 3 | - 2 8 ,
n phan tir khong chua 5 phan h i nao nguyen to nhau tung doi mot Vay n > 217 Bay gia ta chung minh rang moi tap hop con chua 217 phan tu cua S deu c6 5 phan tu nguyen to cung nhau tung doi mot chung minh nay dua vao nguyen ly Dirichlet Goi A la tap hgp con cua S c6 |A| >217 Ta djnh nghla Bj la tap hgp gom 1 va tat ca cac so nguyen to ciia S Ta c6 B^ =60 Ta djnh nghla tie'p:
Suy ra tap hgp S \^ u B2 u B3 u B4 u B5 u B^ c6 280 - 88 = 192 phan tu Vi A
CO it nha't 217 phan tir nen c6 it nha't 217 - 192 = 25 it nha't 5 phan tu cua A nam
trong mot tap hop Bj nghla la c6 it nha't 5 phan tu nguyen tociing nhau
^dilO.{Putnam2005)ChoS = {(a,b)\a = l,2, ,n,b = l2,3}
Mot nuac ca cua S la mot duang gap khdc tao bai cdc doan thang not cdc cap diem p j ,P2 ,-,P3n trong day diem dinh nghia nhu sau:
(i) P i S S , (ii) Pj va p^^i each nhau mot dan vi, vai moi 1 < i < 3n, (Hi) vai moi peS ton tai duy nhat i sao cho Pj = p
Hoi CO bao nhieu nuac ca khai ddu tit (1,1) va ket thiic tai (n, 1)?
^di 1 1 Cho nvdkld cdc so'nguyen duang, n > 3,^ < k < n Cho n diem trong
^dt phdng sao cho bat ky 3 diem nao cUng khong cung 0 tren mot duang thang
G»d su moi diem da cho deu not vai it nhat k diem khdc bai cdc doan thang
Cftim^ minh rang ton tai 3 doan thanh mot tam gidc •
7 5
Trang 39Dili iiititir^ i/i!< ^ioi 1 Dun to lur/t - rai rac, Nguyen Van Thong
L a i giai
V i n > 3,i<> ^ , vay k > 2 Vay trong so n diem ciia mat phang ton tai 2 diem
V i va V 2 noi v a i nhau bai mot doan thang Xet cac diem con lai goi A la tap
hop cac diem khac vai V 2 va noi v o i va B la tap hop cac d i e m khac v o i V j
va noi voi V 2, thi |A| > k - 1,|B| > k - 1 , vay
n - 2 > A u B = A + B - A n B > 2 K - 2 - A n B suy ra A n B > 2 k - n > 0 Vay ton tai mot diem V 3 noi v o i V j va V 2 de tao
thanh mot tarn giac
^di 12 ( I r a n 1999) Cho r j ,r2, ,r„ Id cdc sothuc Chung mink rang ton tai mot
^di J3 Mot cuqc hoi thdo todn hgc qui tu 1990 nhd todn hgc tren todn the giai
cho biei cu rnoi nhd todn hoc deu dd c6 dip Idm vice chung vai 1327 nhd todn
hoc khdc tham du hoi thdo Chung minh rang ta c6 the tint dugc 4 nhd todn hoc
titngdoi mot dd Idm viec chung vai nhau 1 , r ; u •» , ,^
L o i giai < , x , t '
M o i nha toan hoc duoc xem la mot diem H a i nha toan hoc da l a m v i f c
chung v o i nhau xem 2 diem dugc noi v o i nhau boi mot doan thiing L y luan
t u o n g t u n h u bai toan 12, goi V j va V 2 la 2 diem dugc noi v a i nhau Ggi A la
tap hgp cac diem noi vai V j va B la tap hgp cac diem no'i vai V2 Liic do: '
A n B | = |A| + |B|-|AuB|>2xl326-664>0 • i v ' - v - 3 « '
nghia la ta c6 the t i m dugc nha toan hoc V 3 da l a m viec chung v a i V j va V2
Ggi C la tap hgp cac diem dugc noi v o i V3 n h u n g khong noi v o i v^ va V2 Ta
CO |C|>1325, vay 1998> ( A n B ) n C = A n B + C - A n B n C
J'A
n g h i a l a A n B n C > A n B | + |C|-1998>664 + 1325-1998 = l > 0
Vay A n B n C ? t 0 va ta t i m dugc v^ e A n B n C Tap hgp gom 4 nha toan
hoc V i , V 2 , V 3 , V 4 da lam viec voi nhau hj-ng doi mot va do la dpcm
^di 14 (Putnam, 2005) Cho Sn Id tap hap moi hodn vi cua tap hap sol, 2,
n vai TieSn, dat 0(71) = 1 neii n Id mot hodn vi chan vd a(7t) = - l neii TT Id
mot hodn vi le Cung vay, ggi v(n)ld so cdc diem codinh hod hodn vi n
Chung minh rhng: Y - ^ ^ ^ = (-1]"+! _iL_
76
Cty TNHH MTV DWH Khang Vi$t
L o i giai Truoc het chiing ta ap d u n g nguyen ly "chua - chiia trong": ggi f la ham so
i c dinh tren tap hgp cac tap hgp con S ciia (1, n) thi
f(S)=I(-l)l"HS|^f(U) : : ^ >
T a S U a T
Trong t r u a n g hgp nay ta lay f(S) la tong ciia cac a(7t) v o i m g i hoan v i 7t c6 diing tap hgp cac diem co djnh la S Vay ^ f(U) = 1 neu |T| > n - 1 va 0 trong cac truang hgp khac, vay
f(S) = ( - l ) " ' P l ( l - n + S )
Tong can tinh dugc vie't n h u sau bang each ggp cac tap h g p diem co d j n h
i=o '"•"•^ i=o i=o '"^-^
IS = O - X ( - I ) " - ^ ^ - ; A = ( - I ) ,n+l
<6di 17.Xdc djnh xem co ton tai hay khong 2 tap hap vo han AvdB cdc diem trong mat phang thoa man cdc dim kipi sau day:
a) Khong co 3 diem nao cua A^jBthdng hang Khoang each nhd nhatgida 2
diem phdn bi^t cua A u B Idl
b) Ton tai 1 diem cua A nam ben trong 1 tam gidc co 3 dinh thugc B vd ton t^i 1 diem cua B nam ben trong 1 tam gidc co 3 dinh thugc A
L o i giai « •'> '>^' Gia su rang ton tai mot c^p tap hgp A , B thoa man dieu kien da cho Chiing
ta bat dau t i m m o t tap hgp 5 diem trong A ma bao h i n h loi ciia chiing (tap hgp loi be nhat chiia 5 diem do) khong chiia mot diem nao khac ciia A Tap hgp
n h u vay dugc xay d y n g n h u sau:
Cho bat k i P e A lay P lam tam ve cac d u o n g tron ban k i n h Ion dan cho den
khi d i qua I d i e m hay nhieu diem ciia A\{P} V i cac diem ciia A co khoang
each giiia chiing it nhat la 1 nen co the t i m dugc 4 diem Q, R, S, T sao cho bao hinh loi ciia P, Q, R, S, T khong chiia phan t i i nao khac ciia A
Gia sii il? la mot ngu giac Liic do la hoi ciia 3 tam giac rai nhau Theo gia thiet m o i tam giac chiia 1 diem ciia B n h u vay vol 3 tam giac ta co 1 tam giac tao bai 3 diem ciia B M a t khac theo gia thiet tam giac nay chiia 1 diem ciia A,
n h u vay mau thuan v o i dinh nghla ciia dl
Gia sii 9? la m o t t i i giac n h u vay thi trong so 5 diem P, a S, T co 4 diem
la dinh ciia 9? con 1 diem 6 ben trong 9? Diem nay tao v o i 4 dinh thanh 4 tam giac, trong 4 tam giac do co 4 diem ciia B, chiing tao thanh 2 tam giac rai nhau trong m o i tam giac chiia 1 diem ciia A N h u vay trong 9? co den 2 diem ciia A trai v o i d i n h nghla ciia 9? ,
Trang 40Bdi duong hoc sink gioi Todn to hop - rai rac, Nguyen Van Thong
Gia su- i)? la mot tam giac Nhu vay trong 9? c6 2 diem cua A, voi 2 diem
nay ta tao dugc 5 tam giac roi nhau c6 dinh la cac diem cua A Trong 5 tam
giac nay c6 5 diem ciia B va chiing tao thanh 3 tam giac trong moi tam giac c6 1
diem cua A vay trong 0? c6 3 diem ciia A trai voi dinh nghia cua 9? N h u vay
truong hop nao cung dan den v6 ly Vay cau tra lai ciia bai toan la khong
<6di iS.(HSGQG 2005, Bai 3)
Trong mat phang, cho bat giac loi AiA2A3A4A5AgA7A8 md khong c6 ba
dttcmg cheo ndo cua no cat nhau tai mot diem Ta ggi moi giao diem cua hai
duang cheo cua bat giac Id mot nut Xet cac ticgidc loi md moi tit giac deu c6 cd
boh dinh Id dinh cua bat giac da cho Ta goi moi tit giac nhu vay Id tiigidc con
Hay tim songuyen n nhd nhd't c6 tinh chat: c6 the to mdu n nut sao cho von
moi i, k e {1, 2, 3, 4, 5, 6, 7, 8} vd i ^k, neu kihieu s(i, k) Id so tugidc con nhdn
Ai, Ak Idm dinh vd dong thai c6 giao diem hai duong cheo Id mot nut da duac
to mdu thi tat cd cdcgid trj s(i, k) deu bang nhau
•-•r^ '-"-"Vi-'-i^- Loi giai •
+ Goi n la so'nguyen nho nha't thoa bai toan
Ta c6: s(i, k) = s (1, 2) voi moi i, k E {1, 2, 3, 4, 5, 6, 7, 8} va i ^ k
+ Do mot niit tuong ung voi c\p dinh nen:
, nC|=^s(i,j) = C8.s(l,2)=>3n = 14s(l,2]^n chia het cho 14 T u do: n > 14
+ Vai mpt hinh lap phuong c6 the ghi tai moi dinh mot so' chpn trong tap {1,
2, 3, 4, 5, 6, 7, 8}, hai dinh khac nhau ghi lai hai so'khac nhau
+ Moi canh hinh lap phuong c6 the tuong ung voi diing 3 canh song song voi no
+ Moi duong cheo ciia mat hinh lap phuong c6 the tuong ung voi diing 3 duong
cheo ciia mat nam cung trong mat chiia no hoac trong mat doi di?n voi no
+ Moi duong cheo (chinh) ciia hinh lap phuong c6 the tuong ung voi diing 3
duong cheo (chinh) con lai
(^di 19.(USAM0 2001, Bai 1)
Co 8 cdi hgp, moi hop chita 6 trai banh Tim son nhd nhat sao cho moi trai
banh tuy y deu duac to mot trong n mdu thoa man dong thai hai dim kien saw
II Trong moi hop, khong cd hai banh ndo duac to cung mot mdu
21 Hai hop bat ki c6 chung khong qua mot mdu
•'.••>''•! Loi giai I
+ Goi xi la so'mau xuat hi^n i Ian i = 1,2, , k (k < n) Ta c6:
Cty TNHH MTV DWH Khang Vi?t
n = + X 2 + + X k (1)
48 = l x i + 2 x 2 + + k X k - ' (2)
^ Goi y la so each chon hai hop khong c6 chung mau nao
Do hai hop bat ky c6 chung khong qua mpt mau nen:
= C2X2 + C3X3 + + C^Xk + y (3)
V i i > l t a c 6 : l - ^ i 4 c f = » ^
2 1 Lay (1) tru - (2) roi cpng voi - (3) ta ducp-c: •
cac giao diem 6 tren duong tuong trung cho cac banh
+ Co diing 8 duong; moi duong chiia diing 6 giao diem va c6 tat ca 23 8iao diem Hai duong bat ki c6 toi dampt diem chung
79