(Khi doi mat v d i cac bai toan ma cac bien la binh dang vdi nhau (nghla la neu doi cho cac bie'n cho nhau thi cdch phat bieu cua ,b^i toan van khong thay doi), chung ta cd the sap thu" tir cac bie'n
^theo mot trat tif tang hoac giam. Viec lam nay tuy cd ve ddn gian, nhifng hieu qua mang lai thi rat ro rang: mot mdt cd the van dung quan he thuf tU de lam cho Idi giai them sac ben, mat khac lai giam di rat nhieu trUdng hdp phai xet trong bai toan.
Chung ta hay bat dau bang mot bai todn ddn gidn:
Vi dy 11. Chiing minh rang trong 50 so nguyen duang phan bi$t bat ky nho han 100 luon cd hai so nguyen to ciing nhau. . ' J
92 Cdc phuctng phdp gidi todn qua cdc ky thi Olympic
Chufng minh. Sap xe'p 50 so' n g u y e n dUOng do theo m o t thu" tir t a n g d a n xi < X2 < • • • < X^Q. Ne'u t r o n g d a y t r e n c6 h a i so' n g u y e n du'cfng l i e n t i e p t h i do c h i n h la h a i so n g u y e n to c i i n g nhau. N e u t r d i l a i , ta c6 X50 ^ x i 4- 2 • 49 ^ 99, m a X50 < 100 n e n d a u b a n g x a y ra. Suy ra d a y da cho la d a y cac so l e l i e n tie'p tCif 1 d e n 99.
DiTdng n h i e n d a y n a y l u o n c6 h a i so' n g u y e n to c u n g n h a u , chang
h a n 1 v a 3. D 6 l a d i e u p h a i chufng m i n h . • V i du 12. Cho A la mot tap hitu han cdc so nguyen thoa man dieu
kien:
(i) so be nhat trong A Id 1, so Idn nhd't trong A Id 100;
(ii) neu x e A\{1} thi ton tai a, b e A de x = a + b.
Hoi tap A CO it nhd't bao nhieu phdn tvl?
Lori giai. G i a su* A = {/ci}^^! w6i \ h ^ ^ • • • K ^ 100.
T h e o g i a t h i e t , v d i m o i m > 1, t o n t a i 1 ^ z < j < m sao cho
km — ki + kj. TO d o :
2kx ^ k2,
2k2 ^ ks,
• • • 1
n-l ^ kn-
Suy r a
2 n- l ^ 2ằ-lfci ^ kn = 100,
hay n ^ 8. V d i n = 8 ta co fcg = 100, suy r a ^7 ^ 50. N e u k-j > 50 thi ke + kr ^ ks = 100, v 6 l y do ke + kj ^ 2^ + 2^ ^ 96. V a y k-r = 50. T i f d n g tif, k^ = 25, suy ra 2ks ^ ke va k4 + k^ ^ ke = 25, tuy n h i e n k^ + k^ ^2^ + 2^ = 24, m a u t h u a n . NhiT v a y , k h o n g c6 tap 8 p h a n tuf n a o thoa m a n de b a i . V d i n = 9, ta k i e m tra trifc t i e p t a p A = { 1 , 2, 4, 8, 16, 24, 25, 50, 100} thoa m a n . V a y t a p A
CO i t n h a t 9 p h a n tuf. • V i e c sap thuf tuf cac b i e n k h o n g c h i hS\i i c h k h i l a m v i e c v d i cdc
so' n g u y e n m a c o n v d i ca cac b i e n thUc. R a t n h i e u b a i t o a n bat d a n g thufc va ciTc t r i diTdc g i a i q u y e t nhanh c h o n g n h d sap thiJ tu"
cac bie'n. Dufdi d a y ta se x e t m o t so v i d u d i e n h i n h .
Nguyen ly cUc tri rdi rac vd mgt so vcng dung 93
V i dy 13 ( M y , 2 0 0 0 ) . Cho a, 6, c Id cdc so thuc khong dm. Chiing
minh rang .,.
a + b + c
L d i giai. D o cac b i e n a, h, c h o a n t o a n b i n h d a n g n e n ta c6 the gia suf a ^ 6 ^ c. K h i do b a t d a n g thiJc c a n chiJng m i n h c6 d a n g :
a + b + c - 3 \ / ^ ^ 3 { y / ^ - ( 1 ) D a t f{b) = a + b + c - ?,\/abc - 3 ( ^ c - , 6 G [a, c]. D 6 thay f"{b) > 0 n e n f{b) la m o t h a m l o i , suy ra g i a t r i I d n nhat cua f(b) se d a t du'dc t a i b i e n : k h i b = a hoac b = c. Do v a i t r o cua a va c t r o n g b a t d a n g thiJc (1) la nhu" nhau n e n ta c h i c a n chufng m i n h trong t r i f d n g h d p a = 6 la d u . K h i do (1) c6 the v i e t l a i t h a n h :
a + 2c + 3\/a?c ^ 6y/ac,
nhi/ng b a t d a n g thiJc n a y l u o n d i i n g theo A M - G M . • Nhan xet. M o t each h i n h a n h , b a t d a n g thiJc t r e n cho ta m o t
" c h a n t r e n " c u a d a i lu'dng:
K h o n g m a y k h d k h a n , c h u n g ta c6 the chiJng m i n h dufdc " c h a n d i r d i " c u a n o l a :
^ m i n | ( v ^ - v ^ ) ' , [ V b - ^ y , ( V ^ - v ^ ) ^ | .
D a y c h i n h l a t r i f d n g h d p r i e n g c u a b a i t o a n M O S P 2000: Cho Qi, 02, . . . , a „ /d cdc so thvCc khong dm. Chiing minh bat dang thiic sou:
+ 02 H + a „
n - <yaia2 •• •art'^\n ( - V / O T ) ^ ) •
M o t each tir n h i e n , l i e u ta c6 the m d r o n g b a i t o a n chan t r e n cho n bie'n? C h u n g t o i diT d o a n r a n g ke't qua t d n g quat sau d a y c u n g d i i n g : Vdi moi so thUc khong d m a, ( i = 1, n ) , to co
+ 02 + h a „ ^ n - 1
- ^aia2 • • • a „ ^ m a x { {y/al - •
94 Cdc phumg phdp gidi todn qua cdc ky thi Olympic V i du 14. Cho a i , 02, 03, 04, as Id cdc so thvCc thoa man dieu kien a\-\- -\- a\ a\ a\ 1. Chiing minh rang gid tri nhd nhd't cua
{ai - ttj)^ vdi i 7^ j khong vuat qud — . i' ChiJng minh. Khong mat tinh tdng quat, gia suf a i ^ 0 2 < as ^
0 4 ^ 0 5 . Dat:
X = min{aj — : 1 ^ z < j ^ 5}.
Ta chi xet x > 0. K h i do O i+i - a, ^ x > 0 v d i m o i i = 1, 4. V d i
j > i, ta cd
ttj - Ci = (Oj - a j - i ) + (a^-i - aj_2) H + (ai+i - a,) ^ ( j - i)x.
Do do (flj - Oj)^ ^ ( j - i^x'^. Lay tong hai ve theo i, j ta nhan dUdc
Ve trai cua bS't dang thtfc tren:
E ( j - ^ ) v = x2 E o--^)' = 50x^
con ve' phai:
/ 5 \
E K- a i ) ' = 5 E ^ ? - E " " ' l^i<j^5 i=l \i==l /
^ 5.
Suy ra
50x2 ^ 5_
Vay ^ — . D o la dieu phai chiJng minh. 1 •
N M n xet. M o t cau hoi difdc danh cho ban doc la, — da phai 1 la h^ng so tot nha't cua bai toan tren hay chifa? Ngoai ra, hay giai quye't bai toan tong quat: Vdi Xi {i = I, n) Id cdc so thUc cd tdng binh phuang bang 1, hay tim so duong nhd nhd't C de bat dang thi^c sau duac thoa man:
m i n{ ( a i - a^f : i ^ j } ^ C .
Mguyen ly cUc tri rdi rac vd mqt so ling dung 95
V i du 15. Cho x\, X2, • • •, Xe72 la cdc so thUc doi mot khdc nhau thuQC khodng (0, 1). ChvCng minh rang ton tai mot cap so (xi, Xj)
sao cho: ô 0 < XiXj{xi - Xj) <
2013
Chiing minh. Kh6ng ma't tinh tdng quat, ta cd the gia suf rang:
0 < X i < < • • • < X 6 7 2 < 1-
V d i moi i = I, 671, dat a{i) = Xi^iXi{xi+i - Xi). Theo bat dang thurc A M - G M :
3a{i) = 3xi+iXi{xi+i - Xi) < (x^+i + Xi+iXt + x^){xi+i - Xi) - _ ^ 3
Lay tong theo i, ta diTdc
671 671
i=l i=l
Tiif danh gia nay, chon chi so i de a{i) nhd nhat, thi 671 • 3a(z) < ^ 6 7 2 - < 1,
suy ra a{i) = Xi+iXi{xi+i - Xi) < ^^j^) nhan difdc d i l u phai
chuTng minh. •
Bai tap nghi r .
Bai tap 14. Chiing minh rang, tit 25 so nguyen duong khdc nhau cd the chon ra hai so, sao cho cdc so con lai khong the bang tong hoac hieu cHa hai so viCa chon.
Bai tap 15 (Baltic Way, 2012). Cho a, h, c la cdc so thUc. Chiing
fninh rang V-P ab + bc + ca + max {|a - b\, \b - c c — a } ^l + ^ia + b + c)\
96 Cdc phuang phdp gidi todn qua cdc Icy thi Olympic
B a i tSp 16. Cho a, b, c, d la cdc so thvcc doi mot khdc nhau. Gidi he phucfng trinh: . ;ii> •^luiyrw/.
a — b\ + \a — c\ + \a — d\t = 1
\b — a\ + \b — c\ + \b — d\t = 1
\c — a\x + \c — b\y -\- \c — d\t = 1
\d - a\ + \d - b\y + \d - c\ = I ' ''
B a i t § p 17. Vdi mSi so nguyen n ^ 3, xet menh de S{n) nhu sau:
"Vdi moi so thUc ai, 02, •. •, a„ deu xdy ra bat dang thvtc:
(ai - a2)(ai - 03) • • • (ai - a„) + [a^ - ai)(a2 - 03) • • • (02 - a„) H h (a„ - ai)(a„ - 02) • • • - O n- i ) ^ 0."
ChvCng minh rang S{n) chl dung khi n = 3 hodc n = 5. :6J
B a i tap 18. C/io o i , 02, a„ /a cdc so nguyen duang phdn biet.
ChvCng minh rang
al + al +•••+ al^ \ a i + a2 + • • • + a„).
B a i tap 19 (DiT tuyen I M O , 2001). Goi A = {ay, 02, . . . , 02011) mot day cdc so nguyen duang, m la so cdc day con ba phdn tit {ai, aj, ak) vdi l ^ i < j < k ^ 2001 sao cho aj = 0^ + 1 vd ak = + 1. Xet tat cd cdc day A nhU vay, hay tim gid tri Idn nhdt cua m.
Tai lieu tham khao
[1] M . Aigner, G . M . Ziegler, Proofs from The Books, Springer, 1998.
[2] DiTdng Quoc V i e t , NhCfng tu tudng ca ban an chiia trong Todn
hoc pho thong, Nha xuat ban Giao due, 2013. h.
[3] Dtfdng Quoc V i e t (chu bien). D a m V a n N h i , Gido trinh Dai so sa cap, Nha xuat ban D a i hoc Sir pham, 2007.
Idi gidi mot bdi todn md 97
[4] DircJng Quoc V i e t (chu bien), L e V a n Dinh, Bdi tap Dai so so cap, Phdn mot so nguyen ly ca ban, Nha xuat ban D a i hoc SiTpham. 2012. | [5] N h o m SV K 5 7 D , Chuyen de Dai so sa cap, D a i hoc Sir pham
Ha N p i .
[6] M o t so tai l i e u tren Internet.
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