bồi dưỡng học sinh giỏi toán tổ hợp rời rạcnguyễn văn thông

'^d gido iCu til - Thqc si Todn hoc - Gido vien chuyen Le Quy Don N G U Y I N VAN THONG TOAN Ti llfP; 111 MC I Danh cho hpc sinh chuyen D M NAII I Toan-„^in^ NHA XUAT BAN DAI HOC QUOC GIA HA NQI Cty TNHH MTV DWH Khang Viet mii Maddu 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; Hanh chinh: (04) 39714899: Tona bien tap: (04) 39714897 Fax: (04) 39714899 Tu ve toan roi rac doi rat som O Trung Quo'c, vao thoi nha Chu nguai ta da biet den nhimg hinh vuong than bi Nha toan hpc Pithagore va cac hoc tro cua ong da phat hien nhieu tinh chat ky la ciia cac so M o t ket qua noi tieng cua cac truong phai la ke't qua ma chiing ta goi la dinh l i Pithagore Tuy nhien c6 the noi rang ly Chiu trdch nhi^m xuat ban Gidmdoc- Tong bien tap : thuyet toan roi rac duoc hinh nhu mpt nganh toan hoc chi vao khoang the ky XVII bang mgt loat cong trinh nghien cmi nghiem tiic cua cac nha toan TS P H A M T H j T R A M hoc xuat sac n h u Pascal, Fermat, Leibnitz, Euler Chiing ta nho lai hai bai toan noi tieng toan roi rac sau: Bi^n tap N G Q C LAM CM ban C O N G TY KHANG V I E T Trinh bay bia 5,., •f>:v,i • ' '^'f • x-^g(f(x)) 1.2 A N H V A T A O A N H : ^inh nghla 1.4.1 Gia s u cho M p t anh xa f t u X vao Y la m p t q u y tac cho t i r o n g u n g v a i m o i p h a n t u x e X m p t p h a n tvr xac d i n h y e Y, k y hi?u f(x) m p t v a chi m p t x e X cho y = f (x) 1.4 TfcH A N H X A ' •' CO Chang han anh xa d o n g nhat I x la m p t song anh v o i m p i X I.I.OINHNGHTAANHXA: ^inhnghia Y la m p t song anh hay m p t anh xa m p t do'i m p t t u ^ I'M _ » G p i la tich anh xa f v a anh xa g K i hieu: g.f nghla 1.2.1 ^inhly 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 b p p h a n t i i y y ciia Y The t h i n g u o i ta gpi: ' " 1.4.2 Gia s u cho The t h i f: X ^ Y, g: Y, g: Y ^ Z, h: Z ^ T 3'- W ' h(gf) = (hg)f ^ y'.Ji^iich • • f(x) la anh cua x b o i f hay gia t n ciia anh xa f tai d i e m x Ta bao phep n h a n cac anh xa c6 t i n h chat ke't h o p • f ( A ) = { y e Y I t o n tai x e A cho f(x) = y ) la anh ciia A b o i f C h u n g m i n h : Ta CO v o i m p i x e X • f"^ (B) = { X e X I f(x) B la tao anh toan p h a n cua B b o i f (h(gf)) (x) = h(gf(x)) = h (gf(x)) = (hg)((f(x)) = ((hg)f){x) Dac biet v o i b e Y, f"^ ({ b }) = { x e X I f(x) = b } De d o n gian k y h i e u ta viet (b) thay cho f"^ ({ b}) v a g p i la tao anh toan p h a n ciia b b o i f M o i p h a n t u K y h i f u £(A) la m o t d i e u l a m d u n g v i f(A) chi c6 nghla k h i A e X R6 rang ta CO f ( ) = v o i m o i f Ta c h u n g m i n h de dang cac quan he: • v» • - s:.v, ^.T ,inur- gf = l x v a f g = l y ^inh nghla 1.4.4 A n h xa f: X 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') f(x'), hay v o i m p i y e Y c6 nhieu nhat m p t x e X cho y = f(x) N g u o i ta 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 nghla 1.3.2 Ta bao m p t anh xa f: X ^ Y la m p t 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 cho y = f(x) N g u a i ta g p i m p t toan anh f: X ^ Y la m p t anh xa t u X len Y ' D i n h l y sau day cho ta biet k h i nao m p t anh xa c6 anh xa ngirpc 1.3 D O N A N H - TOAN A N H - S O N G ANH: ^inh 1.4.8 Gia s i i f: X - » Y v a g: Y - » X la hai anh xa cho Tir d i n h nghia ta suy f cung la m p t anh xa ngupc ciia g B c f ( r ^ (B)) v o i m p i b p phan B ciia Y keo theo q u a n h$ x = x', hay x ^ x ' keo theo f(x) C h i i 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 The t h i g g p i la m p t anh xa ngupc ciia f A c f"-^ (f(A)) v o i m p i b p phan A ciia X ^inh • D o d o ta k y h i e u h(gf) = (hg)f bang h g f va g p i la tich ciia ba anh xa f, g, h 'i>inhnghla x e f"^ (b) g p i la m o t tao anh ciia b b o i f Y CO m p t anh xa ngupc k h i v a chi k h i f la m p t song anh Chung mink G i a sii- f c6 m p t anh xa ngupc g: Y -> X Theo d i n h nghia ta c6: gf = I x v a fg = ly Tucla: g(f(x)) = x v o i m p i x - Xetquanhe: f(x) = f(x') Tasuyra x = g(f(x)) = g(f(x')) = x' , v , , Vay f la m p t d o n anh Bay g i o gia s u y la m p t p h a n t u t i i y y ciia Y D a t x = g(y) e X t r o n g d a n g t h i i c f(g(y)) = y, ta dupe y = f(x) Vay f la m p t toan anh Dao lai, gia s u f la m p t song anh Q u y tac cho h r o n g 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 f X va ta thay Boi duong hQc sink gidi Todn tohipp - rai rac, Nguyen gf = I x Van Thdnjj Cty TNHH va fg = ly g(y) = X, cho f(x) = y g H a i tap Y T h e t h i g = g' ^ T u do: gf = I x va fg'= ly ' ' i s it; = , ' (y) bang i ; ; ^ (f''^)"^ Vay / I I \ / v a i tap h i i u han) V m g t khoang (a, b) bat ky: s u t u a n g l i n g c6 the thuc hien bang Jib Hinh 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 t h i 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 = Ix 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 k h a i n i e m true tiep m o rgng khai n i e m so' t u nhien Luc l u g n g ciia m g t tap A t h u a n g d u g c k i hieu I A I (hay cardA) C h u n g m i n h Ta c6: (g-' / K h i hai tap h g p t u o n g d u o n g nhau, ta bao rang c h i i n g c i i n g m g t luc l u g n g nen f = Ji^ qua Cho hai song anh f: X ^ Y v a g: Y - » Z The t h i gf: X -> Z la m g t song anh (gO( I I \ I \ • phep v i t u ( H i n h 1) y ^ f - l (y) CO tu (0,1) t u a n g d u o n g v a i tap cac d i e m (y) va d o d o n g u a i ta k y h i e u anh xa Ta S T a p cac d i e m khoang D o l a m d u n g n g u a i ta cung k y hieu phan t u d u y nha't x ciia V i f la anh xa n g u g c ciia t u o n g d u o n g v i c6 't ciia n o (dieu k h o n g the xay d o i « f"^ 2n, (n = l , , 3, ) d u a n g v a i m g t b g phan thuc s u y ^ X , v o l X la p h a n ttr d u y nhalt ciia f"^ (y) la anh xa n g u g c ciia f, bang n, } va B = { 2, 4, 6, vay m g t tap v han c6 the t u a n g ' N h u vay neu f: X ^ Y c6 anh xa ngugc t h i anh xa n g u g c la d u y nha't, xac dinhboi h a n c i i n g so'lugng t h i t u o n g d u o n g D i e u dang chii y day la B c A: ' g = gly = g(fg') = (gOg' = l^g' = g'- ^ Vi^t the thiet lap phep t u o n g d u o n g l i n g : n ; f-ig-^=(gf)-i N e u m g t tap A t u o n g d u o n g v o i m g t bg phan ciia m g t tap B, n h u n g k h o n g t u o n g 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 v a n de t u nhien ra: neu m g t tap A t u o n g d u o n g v a i m g t bo phan ciia tap B va n g u g c lai tap B c i i n g t u o n g d u a n g v a i m o t b g phan ciia A , t h i eo §2 L l J C Ll/ONG CUA CAC TAP HOP the n o i gi ve h a i tap ay? D i n h ly sau day giai dap v a n de 2.1.1 ^inh ly (Cantor- Berntein) N e u tap A t u a n g d u o n g v a i m g t bg phan 2.1 TAP HOP TUONG O JONG: K h i ta d e m Ian l u g t cac phan t u ciia mgt tap A, c6 the xay hai t r u a n g hgp: T a i m g t liic nao do, ta d e m d u g c het cac p h a n t i i ciia A T r o n g t r u o n g h g p nay, tap A la h i i u han v a so cuoi c i i n g d a d e m t a i cho ta biet so l u g n g phantuciiaA ly, •' >vi M a i m a i v a n n h i i n g phan t i i ciia A chua d e m t o i T r o n g t r u a n g h g p nay, tap A la v han Ta n o i 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 g i i i a hai tap h g p ay c6 the thiet lap m g t phep t u a n g l i n g - (tiic la c6 the anh xa - tap len tap kia) ' ciia tap B v a n g u g c lai tap B c i i n g t u o n g d u o n g v a i m g t bg p h a n ciia tap A t h i hai tap t u a n g d u a n g N h u vay, neu tap A t u o n g d u o n g v o i m g t 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 < I B I ) N o i each khac l A I < I B I , IBI < l A I Chung mink l A I = IBI theo gia thiet c6 m g t phep t u o n g li'ng - f giiia cac p h a n 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 Hinh 11 BSi duang htfc sink gioi Todn tohgrp - rcri r^c, Nguyen Vin C t y TNHH Thdng M T V DWH Khang Vift mpt p h e p tirong ung 1- g giiia cac p h a n t u cua B vai cac phan t u ciia mpt bp phan ctia A s'< < ^ii ; ' ; ; Ta qui uoc gpi mpt phan t u x la "ho" cua phan t u 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 phep tuong ung f) hoac neu xe B, y e A va y = g(x) (y ung vdi x phep tuong ung g) Vi du;Theo v i d\ (a myc truoc), t a p |2, 4, 6, 2n, } la dem dupe Ta cung thay rang cac t a p { 3, 6, , n , ), { , 4, , 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, - , n , - n , Mpt phan t u x (thupc A hay B) gpi la "to tien" ciia mpt phan t u y, neu c6 mpt day XI, X2, XK cho x la bo cua xi, xi la bo cua X2, X2 la bo ciia X3, XK la bo ciia y Chang han 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 lam ba tap con: Ac gom tat ca cac phan t u 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 t u ciia A c6 v6 so' to tien Dong thoi ta cung theo each chia B lam ba tap: Be, Bi, B~ Sau ta xac dinh mpt phep tuong ung cp nhu sau giua cac phan t u ciia A voi eae phan t u ciia B: voi moi phan t u thupc Ac hay A - thi cho ung eiia no, vai moi phan t u thupc A i t h i cho ung bo ciia no (i r? spi;:; Viec luc lupng dem dupe la lue lupng be nha't eiia cac t a p v6 han dupe xac De thay rang (p la mpt phep tuong ung - giua A va B That vay, phep tuong ung do, voi moi x e A d l nhien ung mpt phan t u nhat y e B Ngupc lai lay mpt phan t u x nao thupc A i ; neu y e Bi thi bo ciia no thupc Ac, nghia la y la eon ciia mpt phan A i ; neu y e Bi thi bo ciia no thupc Ac, nghia la y la ciia mpt phan t u x nao thupc Ac; neu y e B - thi bo' ciia no thupc Aoo, nghia la y la ciia mpt phan t u x nao thupc A - Thanh t h u moi phan t u y e B ung voi mpt phan t u nhat x e A phep tuong ling (p Vay (p la mpt phep tuong l i n g 1-1 giua A va B Tir dinh ly ta suy rang cho truoc hai tap A , B, chi c6 the xay mpt truong hop: •j; lAI = IBI l A I < IBI IBI < l A I • K''^' ->niVx F] Chung minh; That vay, cho M la mpt tap v6 h^n Ta hay lay mpt phan t u bat ky, M , roi mpt phan t u a2 e M \, roi mpt phan t u as e M \, 32), v.v Vi M v6 h?n, nen dieu 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 t a p dem dupe thi phai la h i m 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 t u dau tien ciia B ma ta g a p day { , , , } , a^^ la phan t u t h u hai ciia B day so'do, v.v Neu 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 k e o dai v6 tan thi B = ( ,a^^ } la dem dupe .XA& • A khong tuong duong voi B, hoac bo phan nao ciia B, va B cung khong la t a p dem dupe ^* " Chung mink Cho mpt day tap dehi dupe: A i , A2, A3 Ta hay viet eae phan tu eiia moi tap ay mpt day va xep dat cac day mpt bang nhu duoi day R6 rang tren moi duong cheo (c6 miii ten) chi eo mpt so hiiu han phan tu: tren chung tren duong cheo thii n chi c6 n phan tu (eae apq ma p + q = n +1) Ai A3 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 t u nhien: N * = { 1, 2, 3, n, } Lue lupng eiia tap 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 t u ciia no mpt day v6 han: a i , a , a , a n , ^ ^11 ^ ^12 >^^13 ^332 ^ ^^ ^14 ^23 324 ^33 334 Veiy ta CO the Ian lupt danh so cac phan tu" tren duong eheo thii nhat, roi den duong cheo t h u hai, v.v ^inh ly 2.2.4 Khi them mpt tap hpp hixu han hay dem dupe vao mot tap v6 hcin ' ' ^inh ly 2.2.3 H p p ciia mpt hp huu han hay dem dupe t a p dem dupe cung tuong duong voi bp phan nao ciia A 12 ' duong cheo thu nhat chi c6 a n , tren duong eheo thu hai chi c6: 821,812, v.v noi ^ J"r?>v ^iyjy^ (;• 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 M , ta khong lam thay doi luc lupng ciia t^p M 13 Cty TNHH MTV D W H Khang Viet Boi duang hoc sink gidi Todn tohgrp - rcri r^c, Nguyen Van Thong Chung minh:Cho N la tap thu dugc 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 chi can lap mot phep tuong ung 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 lap dugc voi cac phan t u ciia mot tap dem dugc la dem dugc Noi ro hon, cho A = day CO dang (a^^, {31,82,33, } HJ^ , , ; ^ la mot tap dem dugc, S la tap tat ca cac ) m la mgt so' t u 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 2.3 Ll/C LUQNG CONTINUM: Qua cac v i du tren kia, ta da di t u tap cac so' t u nhien den tap cac so' h i m ti, roi den tap cac so dai s6^ moi tap bao gom v6 so phan t u moi so voi cac tap da xet truoc, the ma luc nao ta ciing chi c6 nhiing tap de'm dugc Vay thi c6 tap nao khong de'm dugc khong? ^inh ly 2.3.1 Tap cac so thuc la khong de'm dugc Chung mink V i 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 de'm dugc, nghla la c6 the danh so' day: Xi,x2,X3, ta hay chia doan [0, 1] ba doan bang Trong ba doan phai CO mgt doan khong chua x^: cho doan ay la Ta lai chia ba doan bang Trong ba doan phai c6 mgt doan khong chua X : cho doan ay la A2 Ta lai chia A ba doan bang nhau, v.v Tie'p tuc mai, ta se c6 mgt day doan Aj A2 3> A3 Nen theo Dinh ly 2.2.3 chi can chung minh rang moi Sm la de'm dugc Voi va voi x^ g A ^ V i I A „ I - > (n -> °°) nen la mgt day doan that lai va theo nguyen ly Cantor, phai c6 mgt diem ^ chung cho tat dieu hien nhien v i rang ca cac dogn ay Co' nhien ^ e [0, 1] Vay ^ phai triing voi mgt x^^ nao ' ViS=U-.iS„ (hie = A Ta hay gia thiet dieu diing voi de'm dugc), va chung minh cho S^^^ (Hj^ ,3(2 , , ; ^ , k ) Giiia s[^+i va ro rang c6 su tuong ung 1-1 (aii'ai2'aim'3k),.-n 0, v d i m o i xau X = ( x i , X , , x „ ] , ta xay d u n g f(X) = ( y i y a - y n + i ) n h u sau: §3 C N G D U N G A N H X A D E G I A I ^di todn 3.1 1,1, mpt xau thupe loai B neu no k h o n g ehiia so h a n g l i e n tiep 0, 0, , hoac , , A la a t h i l y c l u p n g ciia S (tap cac bp p h a n ciia A ) se d u p e k y h i ? u 2°' M O T SO BAI TO AN ROI R A G hoac = 2a^ V g y S g o m 2" p h a n t u : I S I = 2" - * , A j , A ^ (n > k) la tap h p p tat ca n h i r n g n g u a i q u e n v o i A K h i d o m o i m p t t r o n g cac A, k h o n g bat d a u b a n g so'O T i r d o ta ed d p e m N h g n xet: t u vi|e so sanh l u c l u p n g cae tap h p p , p h u a n g p h a p song anh cd the g i i i p c h i i n g ta d e m so' p h a n t i i eiia m p t tap t h o n g qua s y so sanh l u c l u p n g tap d d v d i m p t t^p khae ma ta da bie't so' p h a n t u eiia n o ,» ^ d i todn 3.3 (Vd dich Ucraina Ggi M la so cac songuyen ^ong CO n chit sol 1996) duong viet h? thap phan c6 2n chit so, vd n chit so ggi N la so tat cd cac soviet phan CO n chit so, chi c6 cac chU sol, so Chiing minh rhngM = N 2,3, vd sochUsol \'^{i'^^p]\m B i N H h? thap hang so chU THUAN 17 Cty TNHHMIV f H VHKhangVift Lai giai Vai moi so' c6 n chiJ' so' gom cac chu so 1, 2, 3, va so' chu so bang so' chu so'2, ta "nhan d o i " so'co 2n chu so'theo quy tac sau: dau tien, hai phien ban cua so' duQc vie't ke so' c6 hai chu so', sau cac chu so n chu so'dau va cac chir s6'4 n chii so'sau dugc doi chu sol, cac chu so n chir so'sau va cac chir so n chu so'dau dugc doi chu so v i du: 1234142 12341421234142 ^ 12121221221112 ^ai todn 3.5 Chiing minh rang vai m, n, k e N h u the^ ta thu dugic mot so' c6 diing n chii so' va n chu so 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 so' c6 n chii so' I v a n chu so' 2, ta cat n chiJ so' dau va n chii so'cuoi roi cpng chung theo cot voi quy tac: 1+1 = 1, + = , + = 3, + = 4, va ta thu dugc mot so' c6 n chii so' gom cac chii so' 1, 2, 3, vai so' chii so' bang so cac so each chgn m + n + l - k s ' t u m + n + l so nen se c6 C|^^n^.i each chgn Cach thii 1212122 Vidu 12121221221112 1221112 '-m+n+l-^m*-n+^m-l'-n+l+-+'-m-k%+k Loi giai Ta de'm so cac l < a i Trong cac tap eon cua tap S = {1, 2, d^ng {ai,a2, ,a„+i}, m + n +1}, de tha'y c6 tap ( l < i < m + l ) aj (cac h i n h t r o n n a m ngoai hoac Ta lai C O bai toan t u o n g t u sau: i T r o n g h i n h t r o n d u o n g k i n h c6 10 d i e m a-2 r tiep xiic ngoai v o i nhau) nen ta c6 -j=- > + 2-j2 ^ , i khoang each giira c h i i n g 'Ijj d u a c h i n h t r o n ban k i n h M u o n vay ta xet h i n h v u o n g M N P Q d o n g t a m i v a i A B C D c6 canh bang N/2 va song song v o i canh A B C D De thay hinh • I ve d u o n g t r o n d o n g t a m va d u o n g k i n h Chia t r o n C O t a m la M , N , P, Q va O, ban k i n h bang thoa m a n y e u cau • sdu diem Chiing minh rang so Chitng minh Chia h i n h chCr nhat da cho n a m h i n h : V i C O sau d i e m n e n theo nguyen l y Dirichlet t o n t a i m o t t r o n g n a m hinh tren, ma h i n h chua i t nhat hai t r o n g so sau d i e m noi tren n C K M Gia s u P la m o t h i n h Dat d(P) = m a x I M N ) , M , N e P, va d a i That vay, t r o n g d u o n g tron t a m O d u o n g k i n h hinh tron da cho c h i n phan (xem h i n h ve: hinh tron d i r o n g k i n h va tam phan bang I I , III, V I I I , IX ma moi phan la - h i n h v a n h khan) R6 rang I c6 d u o n g k i n h bang luon tim dugc hai diem c6 khoang each gim chiing khong Ion Hon \ls Ta d u a vao khai n i e m sau: I- ' Xet chang han h i n h I I A B C D (do la - h i n h v a n h khan) Ta hay t i n h d u o n g k i n h ciia no Co the thay d u o n g k i n h cua I I I la d = A D = BC I'll ij V i D0A~=45" nen: F D d2 = DA2 = DO2 l u o n g d(P) gQi la " d u o n g k i n h " ciia h i n h P De thay ca n a m h i n h d e u c6 d u o n g k i n h - 2DO.OA.COS45'' 5-/2 E } Tir d o suy d^ Thi du: d ( D C K F E ) = CE = K E = CF = D K = N/5 + AO2 \ bang ^[S d ( A B C D ) = A C = N/5, 29 A B C D , DCKFE, K F N M , ^ r- - , (do V2 =1,4142 ) Theo n g u y e n l y D i r i c h l e t t o n tai i t nhat hai l i e m r o i vao m p t t r o n g cac m i e n I , I I , I I I , IX ^1 C O d u o n g k i n h bang 2, cac m i e n I I , IX k h o n g I o n h o n yjs Bai toan d u g c c h u n g m i n h C h u y : Ta c6 bai toan t u o n g t u sau: C O d u o n g k i n h b a n g n h a u va bang d < 2, tir d o T r o n g m p t t a m giac d e u c6 canh bSng lay suy t o n tai hai t r o n g so 10 d i e m da cho m a d i e m t r o n g c h i i n g v o i khoang each < i \ Vay d < NFEQR, Q E D A S Til- d o suy l u o n t i m d u g c d i e m t r o n g so d i e m d a cho c6 khoang ca^'i 17 d i e m t u y y C h u n g m i n h rang t o n t a i h a i , ,, C h u n g m i n h rang t o n tai i t nhat hai d i e m m a N g u g c l a i v a i t o giay h i n h v u o n g A B C D c6 canh a = + 2\/2 ta l u o n cat ^ d i todn Trong hinh chit nhat x4Mt Vi?t V i 16 t a m giac n h o d o chua 17 d i e m nen i t nhat m p t t a m giac chiia d i e m v u o n g n h o n e n c6 m o t h i n h v u o n g n h o chiia i t nhat d i e m t r o n g c h i i n g Gia vuong Khang That vay, ta chia tam giac da cho 16 t a m giac deu n h o c6 canh la — Ifxem hinh ve) • ^'-'^ • ' Ta chia M N P Q h i n h v u o n g b5ng b o i cac d u o n g thang qua tarn sir h i n h DVVII khoang each g i u a c h i i n g nho h a n hoac bang 14 245 CtyTNHH BSi duang hgc sink gioi Todn tohqrp - riri rac, Nguyen Van Thong '^di todn Cho mot hinh vuong vd 13 dudng thang, moi duang thang dm hinh vuong thdnh tiigidc c6 ti so di^ Chung chia minh G p i A B la d u o n g k i n h cua Ai,A2, ,Aioo)-Tac6: ' AiA +AiB>AB = Chung minh G p i d la d u a n g thang chia h i n h v u o n g A B C D t u giac A2A + A2B>AB = CO t i so' d i ^ n tich bang - D u o n g thang d k h o n g the cat canh ke cua A3A + AsB > A B = A100 h i n h v u o n g v i k h i k h o n g tao t u giac Gia s u d cat canh A B va CD AiooA + AiooB > A B = tai M va N k h i no cat d u o n g t r u n g b i n h EF tai I Gia su => ( A i A + A A SAMND = - S B M N C EI = - ^200 •• m i n h r5ng ton tai m o t d i e m tren d u o n g tron ma tong cac khoang each t u no den tat ca n d i e m danh dau > n d u o n g t r u n g b i n h cua h i n h v u o n g theo ty ^ d t todn Tren duomg trdn ban kinh ddnh dau 100 diem Chieng minh ton tai diem tren dudng trdn tnd tong cdc khodng cdch tic no deh tat cd diem ddnh dau Ian han 100 rang , ,, i > ,,, A^0A2 + A^OAg + A^0A4 + A^OAg + / ^ A ( , + K h i suy m i n AfiA^ < = 60° (0 day dat A? t r i i n g v a i A j ) Gia sir A|^OA;^+i = AjOAj 100 i K h i d o A^^OA^i < ° V i OAk < 1, O A k i < 1, AkOAk+i < 60" V: Cty TNHH MTV DWII Boi duang hgc sinh gioi To&n tohfrp — rcri rac, Nguyen Van Tttong N e n t a c o Ai^OAi^+j^ < max | A , A , ^ ^ ] ,OA|^^,jA|^ Khang Viet D o — > = 3.1, theo n g u y e n l y Dirichlet c6 m o t d i e m M nao d o thuoc A B la 71 T u d o theo m o i lien he giua canh diem t r o n g c h u n g cua doan t h ^ n g da chieu x u o n g K h i d o d u o n g thSng d i va goc t r o n g tarn giac AkOAk+i t h i qua M v u o n g goc v o i A B se cat d u o n g t r o n AkAk*i < m a x l O A k , OAk+i} < ^di toan 12 Chung minh rang mgt duang thang cat cd canh cua mot tam D i e u m a u t h u a n v o i AkAk+i > j(hi vd chi no di qua mot dinh cua tam (vi he d i e m A i , A , , Ae thoa m a n yeu cau bai toan) ,, , , Ak+ i , ; , chon qua d i e m thoa m a n yeu cau bai toan, hay n < ^di • toan 10 Trong hinh vuong canh bang lay 100 diem bat ky Chiing minh Gia sir cac d i e m da cho la A i , A , gidc ciia tam giac Ngug-c lai, gia sir c6 m o t d u o n g thang d cat ca canh cua m o t tam giac A B C Ta se c h u n g m i n h rang d phai d i qua m o t d i n h ciia tam giac Gia sir n g u g c lai la d k h o n g d i qua d i n h nao ciia tam giac ca K h i d o d chia mat minh rang c6 it nhai diem nam mot hinh trdn ban kinh bang Chiing ph3ng l a m m i e n , theo n g u y e n ly Dirichlet, ton tai m o t m i e n chira it nhat dinh, k h o n g m a t t i n h t o n g quat la d i n h A va d i n h B K h i d o canh A B n a m hoan Aino Ta d u n g cac hinh toan t r o n g nira m a t p h a n g va k h o n g the cat d dugc, m a u t h u a n v o i gia t r o n CO tam la cac d i e m va ban k i n h bang Tong d i e n tich ciia 100 hinh thiet la d di t r o n la IOOTT Tat ca cac h i n h tron deu nam t r o n g h i n h v u o n g M N P Q c6 giac A B C canh bang 10 chua h i n h v u o n g ( A B C D ) da cho Co cac canh song song v a i cac ^di toan 13 Chung minh rang mot duang thang cat ca canh cua mot tam canh ciia A B C D va each c h u n g ve phia ngoai m o t khoang bang Ta c6 SABCD = loi vd chi no di qua mgt duang 100 V i M N P Q chua tat ca cac h i n h t r o n da ve Tong dien tich cac h i n h t r o n la IOOTI > SMNPQ nen M N P Q c6 d i e m O la d i e m t r o n g c h u n g cua i t nhat h i n h t r o n t r o n g so 100 h i n h t r o n K h i h i n h tron tam O ban k i n h bang chua tam cua h i n h tron noi tren ben no • ^di toan 11 Trong mot hinh vuong c6 canh la chiia mot so^duang trdn Tong dai cua chung la 10 Chiing minh rang ton tai mot dubng thang, md no cat nhat nhung duang trdn Chiing minh x u o n g canh thay rang hinh chieu ciia it tat ca canh cua tam giac A B C Vay d phai d i qua m g t d i n h tam • Chiing mot d u o n g t r o n ban k i n h R la mot doan thSng CO dai 2R V i vay tren canh h i n h v u o n g da chon se CO n h i j n g doan thang chieu tong d o dai la — ( G la chu v i h i n h tron) cheo nao ciia tir giac l o i A B C D t h i c6 t r u o n g h g p xay ra: a) d cat canh ben A B va A D K h i d o d hoac k h o n g d i qua B hoac k h o n g d i qua D , gia sir k h o n g d i qua B K h i d o B va C n a m c i i n g phia v o i d va d o d o d khong cat canh BC b) d cat canh d o i A B va C D V i d k h o n g d i qua d u a n g cheo nao ciia t u giac, xuong vai C N h u vay hoac d k h o n g cat BC hoac k h o n g cat A D Sdt toan 14 Tren mat 2(Ri + R2 + Rn) = — ) O phang cho n duang thang titttg doi mot khong song vai Chung minh rang goc giUa hai duang thang ndo so'do khong Ian 180° Hon i| Chiing ^, , minh Lay tren mat p h a n g m g t d i e m bat ky va ke qua d o cac d u a n g thang song song v a i cac d u o n g thSng da cho C h u n g chia mat p h a n g l a m 2n 180° goc, CO t o n g cac goc bang 360" D o d o ton tai m g t goc k h o n g Ion h a n — — thang ndo duong thang khong c6 hai song song Chiing minh rang ta tim dugc hai ^6i tren md goc giiia chiing nhd han 248 cheo minh That vay, gia sir c6 m g t d u o n g thang d k h o n g d i qua d u o n g fidj toan 15 Tren mat phdng cho ( C i + C2 + Cn = 10, gidc cho nen d k h o n g t h a i d i qua B va D T u o n g t u d k h o n g d o n g t h a i d i qua A va Ta chon m o t canh h i n h v u o n g roi chieu v u o n g goc cac d u o n g t r o n De gidc Chung minh Ro rang mQt d u o n g thang d i qua d i n h tam giac se cat ca canh Tir d o ta thay gia thiet p h a n c h i i n g la sai Dieu c6 nghTa la k h o n g the • duang D duang thang 26° 249 Bin iliiaiii; Chang hill siiih i Ioiiii io htrp - red rac, Nguyen Van Thong Cty minh Ta t i n h tien tat ca d u o n g thang da cho cho c h i i n g ciing d i qua m g t d i e m co d i n h O Ta se duoc d u o n g thang chia goc day d i n h O 14 = 25— 2n K h o n g mat t i n h tong quat gia su a i + + a4n > bi + + b4n Dirichlet de giai m o t l o p cac van de khac N g h i e n c i i u n h i r n g v a n de K h i a i + + a4n > bi + + b4n Tat ca cac doan thang da cho deu d u g c chieu se d a n dat c h i i n g ta den v o i m o t k h a i niem, goi la so'Ramsey u n g V i dai cua m o i doan thang bang nen + bi > D o (ai + x u o n g doan thang c6 dai 2n, v i c h i i n g deu nam t r o n g d u o n g t r o n ban kinh n N e u n h u cac h i n h chieu cua cac doan th3ng da cho len d u o n g thSng k h o n g C O d i e m chung, t h i se c6 a i + + a4n < 2n D o tren p h a i c6 m o t d i e m b i cac d i e m cua i t nha't t r o n g so' cac doan thang da cho chieu len no d u o n g v u o n g goc v o i tai d i e m d o se cat i t nha't hai doan thang da cho ^ d i todn 17 Trong hinh vuong doan CO dai bang Chiing ky doan ndo 12 doan bat ky, moi minh rang ta c6 the dung dugc mot hinh trdn c6 ban kinh bang nam hinh vuong te^ la bai toan t r o n g de t h i I M O lan t h u 6, to chuc tai N g a d o H u n g a r i de n g h i Vidu J ; C o 17 n g u a i trao d o i v o i nhau, m o i n g u a i trao d o i cho tat ca n h i i n g n g u o i lai T r o n g n h i i n g cuoc trao d o i chi c6 van de d u g c thao l u a n M o i cugc n g u o i trao d o i cho n h a u ve c i i n g m o t van de GJ c6 canh bang 10 ke 12 doan thang De bat d a u , ta c u n g p h a n tich m o t bai toan t r o n g k i t h i O l y m p i c toan quo'c da cho va khong c6 diem chung vai bat Ta se giai quye't v a n de sau De l a m d o n gian l o i giai, ta xet hai bai toan |dan gian, n h u n g c6 lien quan de'n bai toan Ion Vi du C h u n g m i n h rang t r o n g n g u a i bat ky, hoac c6 n g u o i q u e n n h a u tirng d o i m o t hoac t u n g d o i m g t k h o n g quen , , j ,, minh Xet m o t d o a n thang A B c6 dai bang Xet Ian can cua doan Vi du Co d i e m dat t r o n g k h o n g gian (khong c6 d i e m nao t h a n g hang, thang A B (tiic tap h o p tat ca cac d i e m c6 khoang each den gan nha't ciia AB k h o n g C O d i e m nao d o n g ph5ng) Co 15 doan thang d u g c n o i t u cac d i e m k h o n g v u g t qua 1) De tha'y Ian can ciia doan A B la h i n h g o m hai h i n h v u o n g n g u o i ta to m a u xanh hoac len cac doan t h i n g C h u n g m i n h rang c6 n h i i n g canh bang 1, c h u n g canh A B , n a m ve hai phia ciia A B (do la A B N M va ABPQ tam giac d u g c to c u n g m a u d cac canh Chiing tren h i n h ve ben) va hai n u a h i n h t r o n n a m ngoai h i n h chCr nhat M N P Q , c6 tarn la A, B va ban k i n h bang L a n can ciia A B n h u vay c6 dien tich la + K Ta c h u n g m i n h v i d u bang n g u y e n H irichlet H i n h the h i e n d i e m d u g c no'i t h a n h doan thSng 15 d o a n t h i n g d u g c to bang m g t t r o n g h a i m a u : hoac xanh Xet d i n h A: Co d o a n thMng c6 d i n h A (AB, A C , A D , A E , AF) d u g c to b a i m g t t r o n g hai m a u xanh hoac D 250 , , Cty TNHH MTV Boi ducntg hoc sink gioi Tnan tohfrp - red r^c, Nguyen Van Thong Theo nguyen ly Dirichlet, mot hai mau, gia su xanh dugc to cho it nhat canh Do tinh doi xung, gia su la AB, AC, A D Bay gio ta xet canh BC, BD, CD Neu mot chung (gia sii BC) dugc to mau xanh, thi chiing ta CO "tam giac xanh" ABC Neu khong c6 canh nao dieu kien nay, ta c6 "tam giac do" BCD DVVf! Khang Vt^t Vay phai can c6 it nhat diem de dam bao su ton tai cua tam giac c6 cac canh dugc to ciing mau, voi dieu kien moi canh chi dugc to mgt hai mau tren Van de dugc thao luan tren da dua chiing ta den mgt khai niem so' moi, 5dugc biet voi ten ggi "so Ramsey" Tap hgp cac hinh bao gom cac tap hiru han diem voi cac canh dugc tao i hai dinh bat ki Nhom - k la nhom c6 dung k dinh N h u vay, nhom - c6 Ta tiep tuc chung minh v i du Coi nguai la diem A, B, C, D, E, F dugc bieu dien nhu Hinh Voi hai diem X, Y bat ky, no'i doan thang XY va to mau xanh neu hai nguai quen inh, nhom - c6 dinh va canh dugc noi tit hai dinh, nhom - c6 hinh mgt tam giac Hinh la nhom - va Hinh la nhom - Vai p, q e N ggi R (p, q) bieu ien so'tu nhien nho nhat "n" cho voi moi canh ciia hinh ta to mgt hai nhau, mau neu hai nguoi khong quen Theo vi du 3, ton tai "tam giac au: xanh hoac thi ton tai "nhom - p xanh" hay "nhom - q do" xanh" hoac "tam giac do" Tuong ung, ton tai nguoi quen doi mot hoac N h u vay theo each bieu dien tren, ta c6 R (3, 3) = Nhirng dang thiic sau doi mot khong quen biet R(p,q) = R(q,p) Bay gia chiing ta chung minh bai toan IMO dugc de cap ban dau tugc suy t u dinh nghla: Chung minh v i du Ta coi 17 nguoi nhu 17 diem A, B, C, va vai hai R(l,q) = l R(2,q)-q diem bat ky, ta noi chiing mot doan thang Doan thang no'i hai diem X, Y dugc to mau xanh (tuang t u do, vang) neu X, Y cung trao doi ve van de I So'R (p, q) dugc ggi la so'Ramsey (tuang t u I I , III) Chu y den dinh A c6 16 canh noi dinh A dugc to mau ^inh ly i : ( d i n h l i Ramsey): Vai tat ca cac so'tu nhien p, q > 2, so R(p,q) luon xanh, do, vang Do 16 = 5.3 + theo nguyen l i Dirichlet, mot mau, gia on tai s; su mau xanh dugc to c6 it nhat + = (canh) Do tinh doi xung, gia su AB, J * Tinh chac chan ciia so Ramsey AC, A D , AE, AF va AC dugc to mau xanh Chii y den dinh lai, c6 15 c^nh ? Su xac dinh ciia gia trj diing ciia R(p, q), p, q dii Idn, vugt qua kha nang dugc no'i t u dinh Neu mgt cac canh, gia su BC dugc to mau xanh thi ta tim kie'm ciia chiing ta Trong chuong nay, chiing ta dua dieu kien cho R(p, CO "tam giac xanh" ABC Trong truong hgp khong c6 canh nao dugc to mau xanh, thi 15 canh phai dugc to mau hoac vang Theo v i du 3, ta c6 dugc "tam giac d o " hoac "tam giac vang" Trong bat ky truong hgp nao, ta cQng c6 mgt tam giac c6 cac canh dugc to ciing mau Dieu c6 nghla la c6 it nhat nguai trao doi ciing mgt van de doi mgt vai Ij ^^' ' ' ^ diem) bai toan Ke't luan ciia bai B%^-^^ toan CO hieu luc hay khong? Nhin vao van de, chii y den hinh vai dinh va 10 doan thang dugc noi t u hai dinh bat ky Neu cac canh AB, BC, CD, DE, EA dugc to \ > thi R (p, q) < ^^'+^.2 (To hgp chap p - ciia tap p + q - 2) (1) mau k, ton tai mau (i = l , k ) va nhom pi cho tat ca cac canh cua nhom pi dugc to lau i : i , ; v , ^ :-f tMian xet: Ta da c6 nhung kien thuc co ban ciia so Ramsey va chung minh R(3, q) < ^ (q^ + q) 'cac moi quan he cua nguyen li Dirichlet va nguyen ly Dirichlet tong quat Djnh l i Tuy nhien dua vao djnh ly 4, ta c6 cong thuc manh hon dugc neu duoi day: C h o q N , R ( , q ) < i (q2 + 3) llpSI voi k > So Ramsey R(pi, p2, pk) la so t u nhien nho nhat n cho voi moi canh dugc tao tir nhom n ta to mgt k mau Mau 1, mau 2, ly : khong kho de chung minh, quy nap tren p + q, ta c6 ke't qua sau: Khi p = 3, bat d^ng thuc (1) tro thanh: p xanh" va cung khong chua "nhom - q do" j ve so Ramsey bieu dien mgt phan nho ciia cac dinh l i sau sac hon va tong quat hon ciia djnh ly Ramsey (2) Cdc bai tap dp dung Chung minh rang voi diem bat ky mgt tam giac deu c^nh don vi, Khi p = q > 3, ta cung c6: - j J - < R (p, q) < 4R(p - 2, p) + :•-'.:•('••' i luon CO hai diem ma khoang each giiia chung khong Ion hon ^ ' v2e Cong thuc tren voi gioi han cho R(p, p) dugc chung minh boi Erdos bang phuong phap xac sua't , Vi du chung ta su dung djnh li thu dugc ke't qua chinh xac ciia R(3, 4) Cho tap C gom n + diem phan biet (n e N) dat tren duong tron don vj, chung minh rang ton tai a, b, e C, a ?t b cho khoang each giita chiing khong |l6n hon 2sin — •r-: Chiing ta biet R(2, 4) = theo (vi du 1) va R(3, 3) = Ca hai so deu chan, theo dinh li 254 255 Tap S g o m d i e m t r o n g i i i n h v u o n g d o n v| C h u n g m i n h rang t o n tai (ii) D i e m ( a i , 32, a n ) t r o n g k h o n g gian K " , v o i n t u n h i e n > d u g c g p i la d i e m phan blot t r o n g tap S ciio di#n tich t a m giac bieu dien b o i diern (Jiem n g u y e n neu tat ca deu nguyen C h u n g m i n h rang v o i tap L n g o m 2" + (Jiem n g u y e n t r o n g M " , t o n tai hai d i e m thupc L n cho t r u n g d i e m ciia k h o n g Ion hon ^ ^ l i n g c u n g la d i e m n g u y e n C h u n g m i n h rang t r o n g m o t tap g o m c6 so', l u o n t i m dug^c so'ma to'ng ty.y 'il;yt;;x c u a c h u n g chia hot cho \' • ;n>il/f »-.;•!- C h u n g m i n h rang ta l u o n c6 hai so n g u y e n p h a n bi^t thupc A cho to'ng Tap A g o m n + phan t u (n e N ), C h u n g m i n h rang t o n tai a, b e A vol a=^b cho n| (a - b) Cho A = jai, a2, 15 Cho A g o m 20 p h a n t u n g u y e n p h a n bi?t chpn tir ca'p so'cpng , 4, 7, (ia c h u n g la 104 ^M;.fM'v(''»'='i.-'!'Cfi'«^H'"^ 16 G p i A la tap g o m d i e m t r o n g mat p h a n g cho k h o n g c6 d i e m nao VLIM], k > la tap g o m 2k + s o t u nhien khac O C h u n g bang hang C h u n g m i n h rang ton tai d i e m thupc A cho n o lap t h a n h m p t 2k+l m i n h rang v o l m o t hoan vj bat ky cua A : '3i2'•••^i2i 2), ton tai it nhat hai nguoi CO cung so'nguoi quen n h o m (gia thuye't su quen c6 t i n h d o l xung) 10 G o i C = j n , r2, rn+i} la tap ciia n + so thuc v o i < ri < , i = l , n + l c h i i n g m i n h rang t o n tai rp, rq t r o n g C, v o i p q cho ' " p - ' " q [...]... u, c6 mgt each phan pho'i n do vat gio'ng nhau cho p nguoi sinh voi so' sinh vien nam t h u hai c6 the se khong eho cau tra loi diing boi vi so' sinh vien n i i nam t h i i hai se dugc tinh hai Ian Chinh v i vay so 'sinh vien trong lop hoac la nix hoac la sinh vien nam thii hai la tong so sinh vien n u va so sinh vien nam t h u hai t r u d i so sinh vien nir nam thii hai Ki thuat giai bai toan de'm nhu... h i sinh k h o n g giai d u g c cau nao HAN a) Tnrcmg hgrp h a i tap h g p : Cho hai tap hop huxi han A va B ta c6 cong thuc A u B = A + B - A n B ' ' 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 ^on Todn vd Ly, 9 hgc sinh. .. gidi hai mon Ly vd Hoa, 10 hgc sinh gidi hai hai mon T^odn vd Hoa Hdi bao nhieu hgc sinh gidi cd ba mon? 30 31 Boi duang hgc sinh gidi Todn to hop - rai rac, Nguyen Van Thong Cty TNHHMTV DWH Khang Viet 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 A u B u C = T \ [ T \ ( A u B u C ) ] Nen so'hoc sinh gioi it nhat mot mon la t... xep 2 hoc sinh n g o i canh hoac d o i d i f n p h a i khac t r i r o n g la: mot cdu veHinh 12 X 6 X 5^ X 4^ X 3^ X 2^ X 1^ =1036800 b) Ghe So' each xep cho mot dethi 1 12 2 11 3 10 4 9 5 8 6 7 12 6 10 5 8 4 6 3 4 2 2 1 tich, hgc Trong 60 tht sinh dvc thi, co 48 tht sinh gidi duoc cdu So 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... = ct5-i= — = 70 : + X2 + + x^, = n , trong do n la mgt so Cl,_,=Cl=15; Cl^s-i = Cg = 70, BU T R 0 TONG Q U A T Midnxet Mgt lop toan rai rac gom 30 nix sinh va 50 sinh vien nam t h u hai H o i eo bao nhieu sinh vien trong lop la n i i 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 =126 nen c6 ca thay 5 +15 + 35 + 70 +126 =... 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? Loi giai V a y so'each xep 2 hoe sinh n g o i d o i d i f n phai khac la: 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 1 2 x 6 x 1 0 x 5 x 8 x 4 x 6 x 3 x 4 x 2 x 2 = 33177600 giai dugc cau So hoc,... -'^^ ' Bieu thuc nay dem mgt Ian cac phan t u chi thugc dung mgt trong ba tap da 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|... giai a « Han -2 so hgc sinh dat diem gioi a Van cung dong 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) Ldi giai Ki hi|u T, L, V, S Ian lugt la tap hgp cac hoc sinh gioi Toan, Vat ly... pho'i k do vat vao m ngan keo |AXB| = |A|.|B 29 Cty TNHH MTV DWH Boi 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 Nguai ChOng minh: ban ddi c6 2 day ghe dot dien nhau, moi day gont c6 6 ghe ta muon xep cho ngoi cho 6 hoc sinh trudng A va 6 hoc sinh truang B trudng ki 2 hgc sinh ndo ngoi c^nh nhau hoac dot di?n nhau tht khdc cho ca hai tap h o p A... trong 14 hoc sinh eo: Chii y: Ta eung eo the xet so' each ehpn m doan ngang thay cho n doan dpc Khi do so duong ngan nhat t u diem (0, 0) toi diem (m, n) se la Cj^^^ N h u vay ta da chung minh dupe bang hinh hpc dang thuc C^^^ = Cj^+n 2.5 MOT S 6 TINH CHAT QUAN TRONG C U A C A C S6 2.5.i.Ne'u 0 < k < m t h i Zi (2) Chicng minh: each Chon A n va Binh roi chon them 4 hoc sinh trong 12 hoc sinh eon lai