intTypePromotion=1
zunia.vn Tuyển sinh 2024 dành cho Gen-Z zunia.vn zunia.vn
ADSENSE

Tuyển tập các phương pháp giải toán qua các kỳ thi Olympic: Phần 1

Chia sẻ: Cô đơn | Ngày: | Loại File: PDF | Số trang:81

703
lượt xem
82
download
 
  Download Vui lòng tải xuống để xem tài liệu đầy đủ

Tài liệu Các phương pháp giải toán qua các kỳ thi Olympic bao gồm các chuyên đề thuộc tất cả các lĩnh vực toán Olympic như: Đại số, giải tích, hình học, số học và tổ hợp với những mức độ chuyên sâu khác nhau. Bên cạnh đó Tài liệu cũng giới thiệu các đề thi và lời giải cùng những bình luận chi tiết các kỳ thi quan trọng nhất về toán của Việt Nam. Mời các bạn tham khảo phần 1 tài liệu.

Chủ đề:
Lưu

Nội dung Text: Tuyển tập các phương pháp giải toán qua các kỳ thi Olympic: Phần 1

  1. CAC PHU'aNG PHAP GIAI TOAN QUA C A C KY THI OLYMPIC NHA XUAT BAN BAI HQC Q U G C GIA T H A N H P H O H 6 C H I MINH - 2 0 1 3
  2. cAc PHi/dNG PHAP GIAI T O A N Q U A LQI NOI D A U C A C Ki T H I O L Y M P I C Trail Nam Dung (Chu bien) Thang 7/2013, Doan hoc sinh Viet Nam dir thi Toan qu6c t^ V6 Quoc Ba Can, LePhiic Lff 2013 tai Colombia da dat thanh tich vang doi voi 3 huy chuang v ^ g va 3 huy chuang bac, trong do c6 02 huy chuang vang cua dai ,0 NHAXUATBAN D A I H O C Q U O C GIA T H A N H P H O H O C H I M I N H dien cac tinh phia Nam la Pham TuSn Huy (hoc sinh lap 11) va CSn Khu pho 6, Phatag Linh Trung, Quan Thu DCfc, TPHCM So 3, Cong trucrng Quoc te, Quan 3, TP Ho Chi Minh frrSn Thanh Trung (hoc sinh lap 12) cua Truang Ph6 thong Nang DT: 38239171 - 38225227 - 38239172 khilu - Dai hoc Qu6c gia TP H6 Chi Minh (DHQG-HCM). Thanh Fax: 38239172 - E-mail: vnuhp@vnuhcm.edu.vn tich nay chac chan se lam nuc long cac em hoc sinh chuyen toan a PHONG PHAT HANH NHA XUAT B A N TP Ho Chi Minh noi rieng va cac tinh phia Nam noi chung. D A I H Q C Q U O C GIA T H A N H P H O H O C H I M I N H De giup cac em dang yeu thich mon toan tiep can vai cac de thi So 3 Cong trucmg Quoc te - Quan 3 - TPHCM DT: 38239170 - 0982920509 - 0913943466 va cac chuyen de Olympic toan hoc, cac giang vien Khoa Toan - Fax: 38239172 - Website: www.nxbdhqghcm.edu.vn Tin h9c, Truang Dai hoc Khoa hoc Tu nhien, DHQG-HCM cung cac cong sir da thirc hien viec bien soan cu6n "CAC PHl/ONG I I A'! track nhiem xudt ban: NGUYEN H O A N G DUNG PHAP GlAl T O A N Q U A C A C K Y T H I OLYMPIC". Noi dung Chiu track nhiem noi dung: cuon sach nay bao gom cac chuyen d8 thuoc tdt ca cac ITnh vuc toan HUYNHBALAN Olympic: Dai s6, Giai tich, Hinh hoc, S6 hoc va T6 hop vai nhung To ckiic ban thdo va chiu track nhiem vd tdc quyen muc dp chuyen sau khac nhau, vi thi, phii hop cho t^t ca cac hoc N H A XUAT BAN DAI HOC QUOC GIA TPHCM sinh chuyen toan. Ben canh do, cuon sach ciing giai thieu cac dl thi Bien tap: va lai giai ciing nhung binh luan chi tilt cac ky thi quan trong nhdt CAO NGHI THUC ve T o ^ cua Viet Nam trong nam qua. Siia ban in: NGUYEN HUYNH De CO dugc cu6n sach day dan vai nhung noi dung phong phu Trinh bay bia: va bo ich nay, nhom tac gia chan thanh cam an sir dong gop nhiet tinh tir cac thay c6 giao, cac ban hoc sinh, sinh vien, cac thanh vien cua diln dan mathscope. Chung toi cung cam an cac d6ng nghiep Ma s6 ISBN: 978-604-73-1976-3 cua Chuang ttinh trong dilm quoc gia phat trien Toan hoc nam So lugng 1.000 cuon; khd 16 x 24 cm. 2010-2020; cac d6ng nghiep Khoa Toan - Tin hoc Truing So dang ky He hoach xuat ban: 1447-2013/CXB/06-82/DHQGTPHCM DHKHTN, DHQG-HCM; dac biet cam an GS. Ngo Bao Chau, GS. Quyet dinh xuat ban so: 194 ngay 30/10/2013 ciia NXB DHQGTPHCM. Le T u k Hoa va TS. Nguyen Thj Le Huong da luon dong vien, c6 Ill lai C ong ty TNHH In va liao bi Hung Phii. \ o p luu chieu quy IV nam 2013. vu va djnh huang cho chung toi. Cam an GS. Dam Thanh San (Pai
  3. hoc Chicago, Hoa Ky) da gu"i cho chung toi file video ve bai toan 3 cua I M O 2013 va ggi y tuang hinh bia cuon sach nay. Chung toi cam an Cong ty c6 phkn giao due Titan da khich le MUC LUC • • ehiing toi hop tac vai Nha xuSt ban Dai hoe Qu6c gia TP H6 Chi Minh d l xuk ban cuon sach nay. ,4 , Cu6i cimg, chung toi cam an sir yeu thieh va say me toan hoc Lcfi n o i d a u ill ciia cac em hoc sinh chuyen toan la nguon dong lire giup chung toi Nhu-ng d i c u t h u vj v l t6'ng P^.(n) = l * - + 2''+ • • + 77^ 'i bien soan cuon sach nay. Trcin Nam Dung 1 ii'i' Chue cac em thanh cong. ' ' ' B o (Ic n a n g l u y thufa va liTng d u n g - * Nhom tac gia Trail Minh Hien, Hoang Dang Thien, Vu Xudn Anh . . . . 19 fit' ..rvu B a t b i e n v a nil'a b a t b i e n t r o n g cac b a i t o a n t r o chcTi Nguyen Thanh Khang 47 C a p so' n g u y e n , c a n n g u y e n t h u y va lifng d u n g Nvuxen Dux Lien 62 N g u y e n l y ci/c t r i rofi r a c v a m o t so' i^ng d u n g DiMng Di'fc Lam 79 X u n g q u a n h so' F i b o n a c c i ! Hoang Minh Qudn 98 LUi g i a i m o t b a i t o a n mnf Vd Quae BdCdn 137 I'• • • DifoTng d o t t r u n g va m o t so' b a i t o a n a p di^ng Trdn Ngoc Thdng 144 D i n h l y C a s e y v a tfng d u n g Nguyen Van Linh 157 M p t so phuTcyng p h a p g i a i b a i t o a n t o n t a i t r o n g t o hgTp Nguyen Tat Thu
  4. M O t e a c h d6'i b i 6 n v a uTng d^ing t r o n g chuTng m i n h ba't d a n g thufc N H O N G O I E U T H U VI V E TONG Nguyen Van Qui va hoc sink Nguyen Qudn Bd Hong . . . 204 i^(n) = i*^ + 2*^ H — ± p f ' : ' : : . D a y s6 = ttn + ( u n ) " , : , j v KieuDinhMinh 232 TranNamDungt i6 ; fi • ' • mi} P h a n hoach nguydn v a phSn hoach tSp h^p LePMcLa 250 M o t hoc s i n h c h u y e n t o a n chac c h a n p h a i b i e t i t n h i e u v e t o n g LoTi g i a i de t h i c h p n H S G t o a n quoc gia n a m 2013 Sk{n) = \'' + 2^+ + N g a y tOr i d p 4, I d p 5, c h a c c h a n c h u n g ,,,,,,,.,,..,„.„,,.. ... 285 ta da tiJfng n g h e c a u c h u y e n ve c a u be Gauss da t i n h t o n g cac so n g u y e n tuf 1 d e n 100 c h i t r o n g n h a y m a t : Lflfi giai d l t h i c h p n dpi t u y e n O l y m p i c t o a n quoc gia nam 2013 1 + 2 + 3 + • • • + 100 = 50 • (1 + 100) = 5050. 0 1 / » p y j^
  5. 2 Cdc phuong phdp gidi todn qua cdc ky thi Olympic Mining dieu thii vi ve tong Pkin) = l*" + 2*" H h n'' B a i vie't nay de cap d e n nhiJng each t i e p c a n khae n h a u de t i n h Nhii v a y . de t i n h t o n g Sk{ii). ta chi c a n t i m d a y so { f „ } sao cho t o n g Ph{n). qua do se gicli t h i e u nhCng tinh c h a t thii v i c u a I\{n). _ = r?/. D i e u nay cd the thufc h i e n b a n g each x e t f „ la m o t C h i i n g ta se g a p l a i nhiTng y tu'dng cua Johann F a u l h a u b e r , Jacob da thufc bac k + 1 theo n va ap d u n g phu'dng p h a p he so' ba't d i n h . B e r n o u l l i , Jacob Jacobi qua each t r i n h bay scJ cap va n g o n ngu" h i e n T a x e t v i du A-= 4. ,^ ^ d a i . B e n c a n h d o , b a i v i e t e u n g g i d i i h i e u m o t so b a i t o a n O l y m p i c k h a i thac c h u de ve t o n g Pk[i}). G i a suf v„ = ATV' + Bn^ + Cn^ + Dn^ + En (ta c d the gia sijf he so tU do b a n g 0). K h i d d d a n g thufc y „ - Vn-\ n ' tUdng dufdng v d i 1. C a c phiftfng phap tinh Pk{n) Arf + Bri^ + Cn^ + Dir + E{n + 1) ' - [.4(/; - l)''^ + B{n - \Y + C(n - if + Din - 1? + En] = n\ N e u nhuf c o n g thu'e da dufde t i m ra nhif (1) hoac (2) t h i v i c e chiJng m i n h la k h o n g k h d k h a n , c h i i n g ta cd the d i j n g m o t p h e p q u y nap So sanh he so' eiia /?.' d h a i ve v d i z = 4, 3, . . . , 1, 0, ta diTdc h e : T o a n hoc ddn g i a n . V d i k = 3, ta cd the cd n h a n x e t l a : ' 5.4 = 1 ;' ' 1^-1 = 1-. -10A + 4B = 0 l ^ + 2^ = 9 - 3 ' ^ = ( l + 2 ) ^ lOA-GB + 3C = 0 1^ + 2^ + 3^ = 3" = 6- = (1 + 2 + 3)1 - 5 . 4 + 4B -3C + 2D = 0 A - B + C - D + E = 0 Tijf do du'a ra du" d o a n : Tii d d g i a i ra diTde A = B = C = D = 0, E = -—. Nhii T 2 ^ • 5 2 3 30 n[ri -t- 1) the: 1^ + 2^ + • • • + 7i' = (1 + 2 + • • • + n)' = , 1 5 1 , 1 3 1 n{n + l){2n + l){3n'+ 3n - 1) V a sau d d ap d u n g p h e p q u y n a p T o a n hoc de chiirng m i n h . T u y ^ ^ 5 2 3 30 30 n h i e n , v d i k bat ky t h i v i e c diT d o a n c o n g thufc la ba't kha t h i . Du^di Phu'dng p h a p nay k h a d d n g i a n ve m a t y t u d n g va c u n g k h a h i e u d a y , ta du'a ra ba hu'dng t i e p c a n k h a c nhau ( d e u d du'di d a n g t h u a t qua k h i A- la cac so' n h d . T u y n h i e n , n e u k I d n t h i v i e c t i n h t o a n t o a n va m a n g tinh t o n g q u a t ) de t i n h Pi,{n). c u n g kha phiJc t a p . T a cd the c a i t i e n phuTdng p h a p n a y de viec tinh t o a n d d n g i a n va h i e u qua h d n nhif sau: ^ , 1.1. Phifcfng p h a p sai p h a n Tru'dc h c t ta du'a ra m o t so d i n h n g h l a va t i n h cha't c i i a t o d n tuf sai p h a n A va toan tuf ngUdc V : » t N h a e l a i phu^dng p h a p tinh t o n g b a n g sai p h a n : D e t i n h t o n g Yl k=i D i n h ngliTa 1. Cho day so { x , , } , day sai phan cua day { x „ } , ky hieu ta t i m day so { r , , } sao eho c/,. - T A . i = in-. K h i d d : Id { A x , , } , ducfc xdc dinh hdi cong thiic Ax„ = x„ — x „ _ i . UA- = (t'l - r„) + (i'2 - t'l) + • • • + (('„ - Vn i) = (V, - To. D j n h nghla 2. Cho day so { x , , } . Ta noi day {?/„} Id nguyen ham ciia day {x,,} neu A?y„ = x „ . Va khi do ta viet y,, = Vx„.
  6. 4 Cdc phuctng phdp gidi todn qua cdc ky thi Olympic Nhang dieu thii v/ ve tong Pfc(n) = 1*= + 2*= H 1- n ' ' De tha'y cAc toin tuf A va V deu la cdc todn tuT tuyen tinh. Viec tinh Pfc(n) se dtfcJc tien hanh thuan IcJi nhd vao tinh tuyen tinh cua = ; ^ n ( n + l ) ( 2 n + l)(3n2 + 3n - 1). V va tinh chat dep de sau: D}nh ly 1. Vdi mgi so nguyen ducfng n , k, ta c6 1.2. Phtfofng p h a p thuTc t r u y h o i V{n{n+l)---{n + k-l)) = • n ( n + 1) • • • {n^k-l){n + k). A; + 1 M o t each tiep can khac de tinh Pk{n) la thie't lap he thufc truy h o i . Chiang minh. Ta c6 Cu the, ta c6 v d i m o i i thi: . / 1 (n - l ) n ( n + 1) • • • ( n + fc - 1) j = k + l 1 n ( n + l ) - - - ( n + fc- l ) ( n + A;) A; + 1 - ( n - l ) n ( n + 1) • • • ( n + A: - 1) Cho i chay tiT 1 den n r o i c p n g l a i , ve theo ve, ta d U d c 1 n{n + l)---{n + k-\) f (n + k) - {n - I) k k-i k+l n{n + \)---{n + k- 1). Suy ra dieu phai chuTng minh. \ TO do ta C O cong thxic truy h o i : De cho gon, ta ky hieu (n'^) = n ( n + 1) ••• ( n + - 1). K h i d6 dinh ly 1 c6 the viet l a i thanh: {n + l)'^'-l-ECl^,PAn) 1 V((n')) = ^ . < n - ) . Pkin) = ^ ^ f ^ . (3) Bay gid ta se van dung dinh ly 1 va tinh tuyen tinh cua V de tinh NhM vay, neu biet cong thiJc tinh Pi{n) \dii < k thi ta se tinh diTdc V { n ' ' ) , tir do tinh diTdc Pk{n). C h i n g han, v d i A; = 4 thi ta phan Pfc(n). Chang han, ta da bie't: tich: = n{n + l ) ( n + 2 ) ( n + 3) + An{n + l ) ( n + 2) + Bn{n + 1) + Cn. s N ri{n + l) n ( n + l)(2n + l) Po(n) = n , Pi(n) = ^ , = • Thay n = - 1 , ta difdc ngay C = - 1 . Thay n = - 2 , ta dUdc B = l va thay n = - 3 , ta dUdc A = - 6 . Tilf d6: A p dung cong thtfc (3) cho = 3, ta c6 V(n^) = V ((n^) - 6(n^) + l{n^) - (n)) P3(„) = (n + 1)^ - 1 - (Po(n) + 4 F i ( n ) + 6P2(n)) ^ , , 1 _ ( n + l)'^ - 1 - n - 2 n ( n + 1) - n ( n + l ) ( 2 n + 1) 4 = - n ( n + l ) ( n + 2 ) ( n + 3 ) ( n + 4) - - n ( n + l ) ( n + 2 ) ( n 4- 3) n2(n + l)2 = : . :y » n j i'' + ^n(n + l)(n + 2 ) - i n ( n + l)
  7. NhQng dieu thii v/ vi long Pfc(n) = l ' ' + 2*= + • • • 4- 6 Cdc phuctng phdp gidi todn qua cdc ky thi Olympic Tie'p tuc dung cong thufc (3) cho k = 4, ta c6 'J'^i^'.. t a t ca cac so' thiTc. D i e u n^y cho p h e p c h i i n g ta x6t dao h a m , cac nghiem khong nguyen... (i^i m jy-M'-Ami ^_{n + \ f - l - (Po(n) + 5 P i ( n ) + IQP^jn) + im{n)) r^i[n) — - Be d a n h d a u d i e u n a y , ta se vie't Qk{x) t h a y v i Pfc(n), v a d i n h nghla cac da thiJc Qkix) b a n g each t r u y h o i n h i f sau: r _{n + if - 1 - n - | n i a + 1) - | n ( n + l ) ( 2 n + 1) 5 Qoix) = x, Qk{x) = . (4) _ n ( n + l ) ( 2 n + l)(3n^ + 3n - 1) X e t cac da thiJc Qk{x), ta t i n h cac dao h a m : , Q\{x) = x + ^, N h i r v a y , dung cong thiJc ( 3 ) , ta c6 the tirng bifdc, t t o g biTdc tinh ra Pk{n). PhU'ctng phap nay dcfn gian ve mat y tudng, nhu'ng ve ky Q'^ix) = x^ + x + ^, ihuat la tiTdng doi phiJc tap v a ta phai thxic h i e n tuan t\i c a c phep tinh Pfc(n) v d i k tH nho den Idn. DiTdi day, chiing ta se x e m xet mot phtfdng phap h i e u qua hdn de tinh Pk{n), hdn nffa phUdng phap nay cho phep tinh Pk+i{n) khi bie't Pk{n), tuTc la ta chi c a n truy hoi Q',{x) = x' + 2x' + x ' - ^ . ve so' hang trtfdc do. Q u a n sat k y m o t c h i i t , ta t h a y cac t h a n h p h a n cua Q'i,{x) r a t g i o n g Qk-\{x), c h i k h a c he so ivC do va m o t he so t y l e . V a thiTc sif t h i 1.3. MOt tinh chat thu vi cua da thiJc Pk{n) va phrftfng ta cd the p h a t b i e u t i n h chat t h i i v i c i i a da t h i i c Qk{x) nh\i sau: phap tich phan Dinh ly 2. Vdi moi so nguyen duang k, ta cd T n / d c he't ta vie't l a i m o t so k e t qua t i n h Pk{n) : I Q'kix) - kQk-iix) = const. Po(n) = n, Chiang minh. T i n h chat d e p de nay cd the c h i i n g m i n h b a n g each „ , , n(n + 1) n sur d i i n g p h i f d n g p h a p quy n a p T o a n hoc. Vdi k = 1, m e n h de d u n g . G i a sijf m e n h de da du'dc chiJng m i n h d e n /: - 1, tuTc la v d i m p i z < fc, ta c d Qi{x) = iQi^i{x) + Ci. i,jAi&y^ N h i c l a i ta cd c o n g t h i i c : * ' , n ( n + l ) ( 2 n + l)(3n2 + 3 n - l ) rv> n Piln) = = 1 1 . ^ ' 30 5 2 3 30 L a ' y dao h a m h a i v e , ta diTdc B a y g i d ta se c6 m o t thay d o i q u a n t r o n g . H a m so Pfc(n) du'dc du'a v a o nhiT h a m so v d i b i e n so k h o n g a m . Nhu'ng b a y g i d , sau k h i Q'^ix) = ix + l)'--^'^CU,Q',{x). . da t h u du'dc no cd d a n g da thiJc, ta cd the m d r p n g d i n h n g h i a cho
  8. 8 Cdc phucfng phdp giai loan qua cdc ky thi Olympic NhCtng dieu thu v/ ve tSng Pk(n) = l'' + 2'' + • • • + n'' 9 A p dung gia thiet quy nap, thay Q'j{x) = jQj^i{x) + Cj v^o d^ng Thay a; = 1, ta difdc cs - 0. TO d6: I thiJc tren, ta diTdc ; lii! , fc-1 fc-i 6 2 12 12 ' , E n^(n+l)'(2n^ + 2 n - l ) 12 fc-i I:M . Neu tiep tuc thiTc hien: = ( x + i)'=-^cr(Q.-i(^) + c.) • . . j=l Q'^ix) = 6Q,{x) + C6 = x' + 3x^ + ^x^ - \x^ + ce, fc-2 ta diWc ^ , ^ x' x^ x^ x^ fc-2 fc-2 Qe{x) = j + - + ^ - j + cex. = (x + 1)'= - 1 - ^ CiQ,{x) + 1- C^c,+i Thay x = 1, ta dtfcJc Ce = ^ . Vay: n''' TV' n = kQk-i{x) + Ck. PM = Qein) = - +- +- - - + 7 2 2 6 42 (Trong bien doi tren, ta da suf dung dang thtfc ^ C ^ + i = = - ^ n ( n + l ) ( 2 n + l ) ( 3 n ' + 6n^ - 3n + 1). D i n h l y dtfcJc chiJng minh. • Chii y . Suf dung each ch^ng minh cua Dinh l y 2, ta c6 the chi^ng C6 difcJc tinh chat nay, ta xay diTng diTdc mot thuat todn hieu minh ket qua tdng qudt sau day di/dc Jacob Bernoulli t i m ra nam qua de tinh Qk{x) (va tuT do Pfc(n)) k h i biet Qk-i{x). Cu t h ^ , ta 1713: Ta dinh nghia cdc so Bn mot each truy hoi nhu sau: biet: m Q'kix) = kQk-d^) + Ck. Tir day lay tich phan, ta dufdc I: j=o Khi do ta c6 Qk{x) = k j Pk-i{x) + Ckx. 1'= + 2*= + • • • + ( n - 1)'= = ^ CUB.n'^^'''. Nhirtig ta khong biet bkng bao nhiSu? D i e u nay khong ddng ngai, chi can thay x = 1 ( v i ta biet Qk{l) = Ffc(l) = 1) vao cong Cdc so Bj duac goi Id cdc so Bernoulli. thtfc tren la ta t i m difcJc ngay Cfc.Vi du, de tinh Qslx), ta c6 Mi Q'.ix) = 5Q,{x) + cs = x' + Ix' + Ix' - l x + c^. 2. C a c tinh chat cua da thuTc Qkix) 2 3 6 Suy ra Ngoai cac tinh chat mang tinh dinh nghia 1^ cong thiJc (4) va cong thurc d dinh ly 2, da thtfc Qkix) con c6 nhieu tinh chat thu v i khac. Difdi day chiing ta xem xet mot so cdc tinh chat do.
  9. 10 Cdc phuang phdp gidi todn qua cdc ky thi Olympic Nhang dieu thii vi ve tong Pfc(n) = 1*= + 2*= + • • • + n'= 11 Dinh ly 3. (1) Neu k Id so le khong nhd han 3 thi Qk{x) chia het Tit cong thiJc nay, de dang chuTng minh bang quy nap dUdc rang Qk{0) = 0, Qk{-1) ^ 0 vdi moi k ^ 3, k le. Lay dao ham hai ve dang thiJc (6), ta duWc Ss*'^*'! 'Ji'a" a'iti (2) Neu k la so chan khong nhd han 2 thi Qk{x) chia het cho 2{k + l)Q'k{x)Hk + l){x + l)'' + {k+l)x' x{x + l){2x + l). - 2 {Cl,Q'k-2i^) + • • • + C',-'Mx)) Chtfng minh. De chtog minh cac tinh chat tren, ta chi can chu-ng - {k + l){2x + l). 1 minh: T£f day de d^ng chiJng minh quy nap difdc • (5fc(0) = 0 = Qj,(0) va Q f c ( - l ) = 0 = vdi k ^ k le; ta do suy ra Qkix) chia he't cho x'^{x + 1)^. , •
  10. f 12 Cdc phuang phdp gidi toan qua cdc ky thi Olympic Nhang dieu thu v/ vi tong Pfc(n) = 1'= + 2*= + • • • + 13 Dieu nay dUdc phat hien dau tien bcli Johann Faulhauber (1580- 3. Cac bai toan Olympic lien quan den tong 1635), mot nha Dai so 1^ ban cua Johannes Kepler va Rene Descartes. Cung nhif phong cdch cua cac nha To^n hoc th6i ky nay, ong chi • Pk{n) , ,,,,, ^.^^^.^ -k^vss.-' ghi ra cac cong thtfc (cho den k = 23) nhulig khong difa ra chiJng minh. Sau nay tinh chat nay dUcJc chuTng minh mot cdch chat che 3.1. De bai bdi Jacobi. Ta phdt bieu tinh chat ma Faulhauber phdt hien dufdc Trong phan nay chung toi gidi thieu mot so bai tap va de thi c6 dufdi dang dinh 1^: Dinh ly 4. (1) Neu k le thi Pk{n) Id da thiic theo N; f lien quan den tong Pfc(n). Cdc bai tap nay cd the cd tinh chat Dai s6' (lien quan den da thiJc Pfc(n), bat dang thuTc), tinh chat So hoc (chia het), Giai tich (danh gia cac tong, gidi han) v^ tham chi To (2) Neu k chdn thi Pk{n) bang 2 n + 1 nhdn vdi mot da thiic theo N. hdp. Cac bai toan nky dtfdc lay iH cdc de thi Olympic cac nu-dc, cung nhif tilf cdc cuon sach Olympic kinh dien. ChuTng minh. Trirdc het, ta chi^ng minh neu k le thi Qk{-x-\) = Qk{x) vdi moi x thUc. Dieu nay c6 the chuTng minh de dang bang Bai toan 1. ChvCng minh rdng vdi moi so nguyen duang k thi tong quy nap va suf dung cong thtfc (6), vdi chu y la vdi k \h thi: jfc _^ 2'= H \-n'' Id mot da thiic bac k + l cua n. Tim he so cua n^'^^ vd cua da thvCc nay. ( - X - 1 + l ) ' ^ ^ - + {-X - \f+' -{k + l)i-x - - 1 + 1) Bai toan 2 (Dai hoc Sir pham Ha Npi, 2010). Gia svC da thvCc P{x) = x'^' + {x + 1)''+' -{k + l)x{x + 1). vdi he so thuc thoa man dieu kien P{n) = l^^^o + 2 2 0 1 ° + • • • + n^^io vdi moi n nguyen duang. Hay tinh P {-\) • NhvC the vdi k le thi Qk{x) Ih hhm doi xu^ng qua diem - 5 . V i vay Bai toan 3. Chiing minh rang neu xi, X2, ..., Xn la cdc so nguyen Qkix) khi khai trien qua luy thufa cua x +1 se chi gom cac luy thiifa duang phan biet thi ch£n. Wi {x + lf = x^ + x + \nen txi day suy ra Qk{x) la da thuTc cua x{x + 1). x\ xl + • • • xl^ {Xi + X'i + • • • ^ Xnf. Bai toan 4 (Rumani, 1999). Chiing minh rang bat dang thiic sau ludn De chiJng minh ve thtf hai cua dinh ly, vdi k cMn ta dat: duac thoa man vdi cdc so nguyen duang phan biet o i , 02, . . . , a„ : Sk{x) - al + al + ---+al^ ^(ai + 02 + • • • + a„). 23; + 1 thi theo dinh l y 3, Sk{x) cung la da thiJc. Tiep theo, suf dung cong Bai toan 5 (Nga, 2000). Day so 02000 thda man dieu kien thiJc (7), ta Viet difdc vdi moin, 1 ^ n ^ 2000 thi al-\-al + --- + al = {ai + a2 + --- + a^f. Chiing minh rang tat cd cdc so hang cua day so deu nguyen. -2{Cl^,Sk-2{x) + --- + C',;lS2{x)). (8) Bai toan 6 (Nga, 2002). Day so OQ, ai, . . . , a„ thoa man dieu kien ao = 0 va 0 ^ a^+i - Uk ^ I vdi moi 0 ^ k ^ n - I . Chang minh Ap dung (8) va ph6p quy nap Toan hoc, ta cung chiJng minh difdc bat dang thiic Sk{-x - 1) = Sk{x) v d i m o i x thiTc va sau do ly luan nhiT d tren de ho^n tat ph6p chuTng minh. • al + al + --- + al^{ai + a2 + --- + anf-
  11. 14 Cdc phuang phdp gidi todn qua cdc ky thi Olympic NhOng dieu thu v/ ve tong Pk{n) = l'' ^ 2'' -\ n'' 15 Bai toan 7 (Nga, 2002). Day so ao, ai, . . . , a„ thoa man dieu kien Bai toan 16. Chiing minh rhng vdi moi so thuc duang a, ta cd ao = 0 va 1 ^ ak+\ vdi moi 0 ^ ^ n — 1. CMng minh bat dang thvCc lim = -. , a\ al + • • • + al^ [ai + a2 + • • • + anf. Bai toan 17. Vdi moi so nguyen n (n > 2), ddt an = 2 + ^ + - • - H - ^ . Bai toan 8 (Rumani, 2001). Cho k vd n i , n^, •. •, rik {ni < 7x2 < Chiing minh rang • • • < rik) Id cdc so nguyen duang le. ChvCng minh rang ; nl-nl + nl nl_-^ + nl^ 2k'^ - I. 2 3 n Bai toan 18. Cdc so 1, 2, ..., n duac viet tren bang. Moi mot phut, Bai toan 9. Cho biet 1^ + 2^ + • • • + = hay chitng minh mot hoc sinh len bang, chon hai so x vd y, xda chiing di vd viet len rang neu Xi, X2, • • •, Xn Id cdc so nguyen duang phdn biet thi ta cd bang so 2x + 2y. Qud trinh nay tiep dien cho den khi tren bang chi {x\ xl + --- + xl) + {xl + xl + --- + xl)^ 2{xl + xl + ... + xlf. cdn Igi mot so. ChvCng minh rang so nay khong nhd han Bai toan 10. Chiing minh rdng neu k la so nguyen duang le vd Bai toan 19. Gid svt x{x+l) • • • {x+k-l) = x''+A^x^'^ + A2X^-^ + n Id so nguyen duang bat ky thi l'' + 2'' + • • • + n'' chia het cho • • • + ^fc-i^;. ChvCng minh rdng ta cd cong thvCc truy hoi: 1 + 2 + • • • + n. ^ (n + l M n - l ) . ^ . ( n - . - + l ) ^ g Bai toan 11 (Bulgaria, 2004). Tim tat cd cdc so nguyen to le p sao cho: p | ( l P - i + 2''-i + - - - + 2004P-^). Bai toan 12 (Hungary-Israel Math competition, 2009). Cho p ^ 2 3.2. Dap so', hufdng din va IM giai torn tat Id so nguyen to. Tim tat cd cdc so nguyen duang k sao cho Sk — l'^ 4- 2^^ + • • • + (p - 1)^ chia het cho p. 1. He so ciia bang c6n he so cua bang \ Bai toan 13. Vdi moi so nguyen duang n, tong 1 + ^ + ... -f- ^ duac 2. Xem dinh ly 3. Ngoai ra, c6 the sur dung mot ly luan khac viet dudi dang vdi Pn, Qn Id cdc so nguyen to cung nhau. nhir sau: Xet p = 2n + 1 la so nguyen to', chiJng minh P(n) chia (a) Chiing minh rang 3 khong chia hit pej. he't cho p. TO do suy ra P {-\) chia he't cho p, chpn p du Idn suy ra P{-\) = 0. Chi tiet xem tai [2]. (b) 71m tat cd cdc so nguyen duang n sao cho 3 chia het Pn- 3. Sap thuf tir x i < X 2 < • • < x„ va chuTng minh quy nap. Chu Bai toan 14. Tim phdn nguyen cua so: S khi do: 2{Xi+X2 + --- + Xn-l) ^{Xn-l)Xn. ^
  12. 16 Cdc phucfng phdp gidi todn qua cdc ky thi Olympic Nhang dieu thu v/ ve tong Pfc(n) = l ' ' + + \-n'' 17 V d i n = 1, ta c6 a\ a\y ra a i = 0 hoac o i = 1. NhiT vay Neu q'^p thi p^ ^pq
  13. 18 Cdc phucfng phdp gi&i todn qua cdc ky thi Olympic [4] Agakhanov N.Kh, Bogdanov I.I, Kozevnikov P.A, Podlipsky BO B E NANG LUY THQA V A O.K, Tereshin D.A, Olympic loan Nga 1993-2006, Nha xuat QNG DUNG .V ban MCCME, Moscow, 2007. (Tieng Nga) , [5] Skliarsky D.O, Chentsov N.N, laglom I.M, Selected Problems Trdn Minh Hien and Theorems of Elementary Mathematics, Dover Publications, va cac hoc sinh Hoang Dang Thi?n, Vu Xuan Anh^ '. New York, 1993. [6] Peter Boyvalenkov, Emil Kolev, Oleg Muskarov, Nikolai Nikolov, Bulgarian Mathematical Competitions 2003-2006, GIL, 2007. [7] Shailesh A Shirali, On Sums of Powers of Integers, Resonance, July 2007. 1. Dinh nghia va tinh chat [8] Beardon A.F, Sums of Powers of Integers, Mathematical Monthly, L)' thuyet nay la hat nhan trong ly thuyet so cong tinh. Day la ket March 1996. qua hay va c6 nhieu iJng dung. Viec hieu ban chat la het siJc quan trong de c6 the suf dung difcfc ket qua nay. [9] Heinrich Dorrie, 100 Great Problems of Elementary Mathemat- Djnh nghla 1. Cho p la so nguyen to, a la so nguyen va a Id so ics, Dover Publications, New York, 1965. tu nhien. Ta noi p° la tide dung cua a. va a duac goi la so mu [10] Ken Wards Mathematics Pages, Series Sums of the Powers of dung cua p trong khai trien cua a neu p'^\avd p°''^^ f a. Khi do ta the Natural Numbers and Bernoulli Numbers, Viet p"" II a vd ky hieu a = Vp{a). Neu a = Vp{a) thi a = p°'k vdi http: //www. tr£ais4mind. coni/personal_Edevelopment/ {k, p) = 1. mathematics/series/sumsBernoulliNumbers.htm. Lay vi du, ta c6 V3{63) = 2 vi 3^ = 9163 va 3^ = 27 j 63. [11] Cac tai Heu ve cac cuoc thi Olympic Toan va mot so' tai lieu Internet khac. R6 rang neu n c6 siT phan tich tieu chuan n = Px^p'^ • Pn" thi khi d6: Vp^{n) = Qfi, Vp^{n) = a2, Vp„{n) = «„. Trong khi lam toan lien quan den n, ta thiTcJng chi lam viec vdi phan tijf "ciTc han", tiJc la u'dc nguyen to p nho nhat cua n. Khi do ta chi quan tam tdi luy thufa cua p trong n, tvlc Vp{n). Noi ve tinh chat cua Vp{n), ta c6 nhan xet sau day: Nh^n xet. Neu p, q la hai so nguyen to khdc nhau thi ta c6 ^Tri/dng THPT Chuyen Quang Trung, Binh PhUdc.
  14. 20 Cdc phuang phdp gidi todn qua cdc ky thi Olympic Bo de ndng luy thiCa vd ling dung 21 Mdnh de 1. Cho a, b Id cdc so nguyen, p Id so nguyen to. Khi do: M^nh de 3. Cho x, y Id hai so nguyen, n Id so nguyen duang le. Xet p Id so nguyen to sao cho: j , , „ ,5;^^, ...; , 1. Mu tv(a) - a, ^ ( 6 ) = /? thi v^{ah) = a + (5. '' ' •'' 2. Neu (p, a) = 1 thi Vp{a) = 0. (n, p\ix + y), p ^> p \ M / 3. Neu Vp{a) = a thi Vp{a^) = ka. Khi do, ta CO Vr ( x " + y " ) = ^;p(x + ^/). 4. Neu Vp{a) ^ a, Vp{b) ^ P va a P thi khi do: Chiirng minh. Do x va y c6 the nguyen am nen suf dung menh de Vp{a + h) = mm{a, (5}. tren ta dUdc: Mot cdch tdng qudt, ta ludn c6 vpix"" - i-yT) =vp{x- i-y)) =^ i;p(x" + y^ = Vp{x + y). Vp{a + b)^ m i n {vp{a), Vp{b)}. Chii y la n la so nguyen dUPng le nen ta m d i c6 sU thay the ( - y ) " bdi - y " . ° 5. Neu n \ thi Vp{n) ^ Vp{m). Dinh ly 1 (Dinh ly ve so mu dung 1 ( L T E ) ) . Cho x,y la hai so ChiJng minh. V i e c chiJng minh cac ke't qua tren la hien nhien. • nguyen {khong nhat thiet nguyen diMng) vd n la so nguyen duang. M$nh de 2. Cho x, y la hai so nguyen vd n la so nguyen duang. Xet p Id so nguyen to le sao cho: Xet p Id so nguyen to sao cho: = p\{x-y), p t X, p \ {n,p) = l, p\{x-y), p \ p \ Khi do, ta CO khi do ta CO ke't qua sau: Vp{x'' - y " ) = Vp{x - y ) + Vp{n). Vp{x''-y'') = vp{x-y). LUu y trong dinh ly tren, khong can gia thiet {p, n ) = 1. Chtfng minh. Chung ta suf dung hang dang thtfc: Chufng minh. D a u tien ta chiifng minh dinh ly doi v d i n = p, tufc la x^-y^ = {x- T/)(a:"-i + x^-^y + x^'Y + •••+ y^-'). can chiJng minh: > M e n h de du'cJc chiJng minh neu chiJng to dUdc: Vp{xP - y") = Vp{x -y) + l. (*) p t (x"-^ + x^'-^y + x " - y + • • • + y^-^). De chufng minh dieu nay, ta se chiJng minh: De chtfng to dieu nay, ta se sur dung gia thie't p \ - y), tiCc x ~ y (mod p). K h i do: p I (x^-^ + x^-^y + • • • + X / - 2 + y"-') (1) „n-3„.2 ,,"-1 va = x"-^ + • X + x"-^. + • • • + X • x"-^ + x' p2 I ( x P - i + x ^ - 2 y + - - - + x/-^+/-^). (2) = nx"-^ ^ 0 (modp), That vay, do x = t/(modp) nen: v i {n, p) = 1, p \ M e n h de dUdc chtfng minh. • + a;P-2y + . . . + = p . xP-^ = 0 (mod p),
  15. 22 Cdc phumg phdp gidi todn qua cdc ky thi turc (1) diTdc chiJng minh. De chuTng minh (2), ta can chii y sau, v i Olympic r de ndng luy thiCa vd ling dung 23 x-y\p nen y = x + kp, v d i k nguyen. K h i d6 v d i m 6 i so nguyen = Vp(^{xP')-{yP'))+a-l = Vp(x -y) + a (sur dung (*) Ian nffa) y'xP-'-' = {x + kpY • x^-'-' = Vp{x-y)+Vp{n). j-i x' + t- (kp) . x'-' +, tit - 1) . {kpf '-^^—^ . x'-^ + Dinh ly difdc chiJng minh hoan to^n. • D i n h ly 2 (Dinh ly ve so mu dung 2 ( L T E ) ) . Cho x, y Id hai so X* + t • {kp) • x'-'] nguyen vd n Id so nguyen duang le. Xet p Id so nguyen to le sao = x^-^ + tkpx"-^ (modp2). cho: V a y ta luon c6: p\{x + y), p \ p \ Khi do, ta cd .i(u. y'x^-'-' = xP-' + tkpx^'^ (mod p^] i = 1, 2, 3, 1. Vpix"" + 2/") = Vp{x + y) + Vp{n). ' • ' Sur dung ke't qua nay ta diTdc: ChuTng m i n h . Y tiTdng chi^ng minh hoan toan gio'ng v d i each ta lam cho dinh ly 1. • x^-^ + xP~^y + ••• + xyP-^ + yP-'^ = xP-' + + kpxP-^) + {xP-^ + 2kpxP-^) D i n h ly 3 (Dinh ly ve so mu dung 3 (LTE) v d i p = 2). Cho x, y Id + •••+ [xP-^ + {p- l)kpxP-^' hai so nguyen le vd x — y\A. Khi do, ta c6 = pxP-' + [1 + 2 + • • • + (p - 1)] kpxP'^ viix"" - y") - V2{x -y) + v^in). kp'xP-' ChuTng m i n h . Ta xet cac trUdng hdp sau: • Neu V2{n) = 0, titc n le. K h i d d : = pxP-^^Q (modp2). y2(x"-y") = y2(2;-2/)+t;2(x"-i+x"-V- • = V2{x-y), V a y (2) dufdc chu-ng minh. Bay gid, v d i n tuy y, ta cd the viet n= v d i {p, b) = l. The thi ta cd Vp{n) = a. K h i d d : v i a;"~^ + x'^^'^y + h 2/"~^ la so le v i la tdng cua n so le, n le. vpix-- yn - v,{{xpy - {ypy) • Neu V2{n) = 1, dat n = 2m, m le. K h i d d : '• = Vp(xP'' - yP"^ (sur dung menh de 2) W2(x"-2/") - V2{x"'-y"')+V2{x"'+y'^) = V2{x-y)-{-V2{x+y). = Vpl{xP-r-{yP-y) 6 day, lap luan dau bang cuoi cung la do menh de 2 va 3. M a t khac, ta l a i cd = Vpi^xP"-' - + 1 (si^ dung (*)) X + y = {x - y) + 2y = 2 (mod 4) , = vpl{xp-r-{yp-y) +i nen V2{x + y) = I. K h i d d , ta cd = Vp (xP'"' - yP""'j +2 (suf dung (*) lin nCTa) V2{x'^ - t/") = V2{x - y ) + 1 = V2{x - y) + V2in).
  16. 24 Cdc phuang phdp gidi todn qua cdc ky thi Olympic Bo de ndng luy thvCa vd ling dung 25 Neu V2{n) = a thi n = 2"6, hie, 1. K h i do: n ( p , n ) = l , ( p , x ) = l , ( p , y ) = l p | ( x - j / ) ^ VpCx" - y " ) = v p { x - y ) n l e ( p , n ) = 1, ( p , x ) = 1, ( p , y ) = 1 p\{x + y) ^ v p i x ' ^ + y ^ ) = v p { x + y ) p ^ 3 , ( p , i ) = l , (p,y) = l p\ y) =4. t ; p ( x " - y " ) = Vj,{x - y ) + V p { n ) -vJ{x'y-{y'Y n l i p ^ 3 , (p,x) = l , (p,y) = l p\{x + y) =^ Up(x" + y") = Vp{x + y) + f p ( n ) = V2 - ^z^") (theo menh de 2) (2,x) = ( 2 , j / ) - l 4 1 (x - y ) =J. i;2(x" - y") = D2(x - y) + i;2(n) n chdn ( 2 , i ) = (2,y) = l 2|(x-y) =^ t ; 2 ( x " - y " ) = •y2(x - y) + t;2(x + y) + t;2(n) - 1 Vi + y^'" ^ 2 (mod 4), Vm = 2, o - 1 nen ta c6 Chiing minh. De chu'ng minh dinh ly nay, ta can den cac nhan xet D i n h ly diTcJc chufng minh hoan toan. • sau day: Dinh ly 4 (Dinh ly ve so m i l dung 4 L T E v d i p = 2). Cho x, y la Nhan xet 1. Neu n la so tu nhien thi n\_x\ \nx\, trong do x Id hat so nguyen le vd n la so nguyen dUcfng chdn. Khi do, ta cd so thUc. v^i^"" - y") = V2{x - y ) + V2{x + y)+ V2{n) - 1 . That vay, dat [x\ a, ta c6 x = a + d v d i 0 ^ d < 1. K h i do ChuTng minh. V i x, y le nen 4 1 (x^ - y^). Do do: ta CO _g ; nxj = n{a + d) = na + \nd\, V2ix'' - yn = V2 {{X')^ - ivT^) = V2{X^ - y^) + V2 Q) do na nguyen. V i ^ 0 nen \_nd\ 0. Suy ra na = n [ x j ^ [nx . , =V2ix-y)+V2{x + y)+V2{n)-l. NhSn xet 2. Vdi moi so tU nhien n vd q (q ^ 0), ta cd q - ^ n. D i n h ly difdc chu'ng minh. • Ldi hay mac phdi khi svt dung cdc cong thvtc tren Id khong kiem That vay, theo nhan xet 1, ta c6 tra dieu ki^n p | ( x ± y ) . 1 n Ta c6 bang torn tat dtfdi day: — q-- n = n. Dinh ly 5 (Dinh ly Legendre). Cho p la so nguyen to vd n la so nguyen duang. Khi do, ta cd Nhain xet 3. Vdi moi sotunhien nvd q {q 0), ta cd n < q il1 + n n n v 1 n - Sp[n) -r + ••• = That vay, viet n = mo + r v d i 0 ^ r < g, ta c6 - = m + - v d i .P. p - 1 r' Q Q vdi Sp{n) la tong tat cd cdc chQ so cua n trong ca so p, tiic la khi 0 ^ - < 1. Do do: n = akP^ + • • • + aip + ao thi: n Spin) = Ofc + afc_i H h a i + OQ. = m.
  17. 26 Cdc phuang phdp gidi todn qua cdc ky thi Olympic Bo de ndng lily thvCa vd ling dung 27 M a t kh^c, tti n — mq + r, ta suy ra n — ( m + l)q + r — q vdti r — q n dung bang , so' cac nhan tuT chia het cho diing bang la so am nen: , ; i\„,.i! M • '> IP'] / n \ D i e u nay c6 nghla la so cac thijfa so nguyen to p c6 trong n < {m+ l)q = + 1 \ .Q. J n n n n + Nhan x e t 4. Trong day n so tU nhien 1, 2, 3, . . . , ? i chi c6 diing P- + .P . Q + • • • + n so tu nhien chia het cho so tU nhien q khdc 0. Gia suT n = a^p'' H 1- aip + CQ la bieu dien cua n trong cd so L9J p. K h i do, vdiO^r
  18. Cdc phuang phdp gidi todn qua cdc ky thi Olympic Bo de ndng luy thiia vd ling dung 29 28 Chu'ng minh. De dang k i e m tra v d i p = 2, 3, 5 deu khong thoa 2. Cac bai toan uTng dung man. X e t p nguyen to Idn hdn 5. Gia suf ton tai x, y thoa de bai. B a i tap 1. Cho p la so nguyen to le. ChvCng minh rang khong ton K h i do vr,{p) - 0. V i 2^ + 3^15 nen suy ra x^^+MS, do 5 nguyen to tai x, y e Z thda man: n6n X : 5. V i y nguyen du'cfng nen y + 1 ^ 2, hay x^+^:5^ = 25. Tuy nhien, ta lai c6 •P + yP = p- [{p-iy. vs{2P + 3") = ^5(2 + 3) + vs{p) = V5i5) + vs{p) = 1. Chtfng minh. Gid suf ton tai x, y thoa man de b ^ i . Nhan thay trong (p - 1)! /p nen: Dieu nay dan den: [{p-iyr/p- 2^ + 3^ /25, TO do suy ra: mau thuan v d i x^+^:25. V a y ta c6 yeu cau bai toan. • • p[{p-i)\Y /p'. B a i tSp 3 ( A n D o , 2002). Cho cdc so nguyen a, b, c thoa man: • Neu X chia het cho p thi do x^ + chia het cho p ( v i ve a\b^, b\c^, c\a^. phai chia het cho p) nen dan den y cung chia het cho p. V i p nguyen to le nen ta c6 x^\p^ va y^'.p^, suy ra Chitng minh rang abc \{a + b + c)^^. ChuTng minh. G o i p la mot Udc nguyen to cua abc. K h i do hoac {x^+y^)\p\ p I a hoac p | b hoac p | c. Gia suf p | a, do a | 6^ nen p | b^, txic ta c6 p I 6. TOdng tu" ta cung chufng minh dUdc p | c. Vay ca ba so a, b, c D o i chieu v d i (1), ta thay mau thuan. deu CO p trong s\X phan tich thanh thijfa so nguyen to. Dat: • Neu x / p , thi tCr dieu k i e n bai toan suy ra y / p . Theo dinh ly Vp{a) = X, Vp{b) = y, Vp{c) = z. Fermat t h i : Do tinh doi xufng, khong mat tinh tdng quat, ta gia suf x = m i n { x , y, z}. x'' + y^ = x + y (mod p) => X + = 0 (mod p ) . Ta can chu'ng minh: Do do theo dinh ly 2 thi: Vp {{a + b + c)^^) ^ Vp{abc) ^ 31 • Vpia + b + c ) ^ Vp{abc). Ta CO Vp{xP + /) = Vp{x + y) + Vp{p) ^ 1 + 1 - 2 =^ (x^ + /) ip', b^ Vp{a) ^ Vj,{b^) => Vp{a) ^ 5 • Vp{b) =^x ^5y, lai mau thuan v d i (1). c^ Vp{b) ^ Vp{c^) =^ Vp{b) ^ 5 • Vp{c) =^y^5z, V a y khong ton tai x, y thoa man de bai. • a^ =4> Vp{c) ^ Vp{a^) Vp{c) ^ 5 • Vp{a) => z ^ 5x. B a i tSp 2. Chitng minh vdi moi so nguyen to p, khong ton tai x. y € Do do: N " sao cho: x + y + z^x + 5z + z = x + 6z^x + 6-bx^ 31x.
  19. 30 Cdc phuang phdp gidi todn qua cdc ky thi Olympic Bo de ndng luy thica vd ling dung 31 V d i ke't qua nay, ta thu dUcJc: B a i tap 5 (Rumani, 2010). Cho so nguyen duang a vd n sao cho tat ca cdc udc nguyen to cua a deu Idn han n. Chiing minh t! ; r Vpia + b + c)^' = 31.Vp{a + b + c) ' ^ 31. m i n {vp{a), Vp{b), Vp{c)} ( a - l ) ( a 2 - l ) . . . ( a " - i - l ) = 31.x ^x + y + z = Vp{a) + Vp{b) + Vp{c) = Vp{abc). chia het cho n\. TO do CO dieu phai chuTng minh. • B a i tSp 4 (Nga, 1996). Cho cdc so nguyen duang a, b, p, n, k tlioa Chtfng minh. V i tat ca cac u^dfc nguyen to cua a deu Idn hdn n man: nen (a, n) = 1. G o i p la u'dc nguyen to cua n\. K h i do ta cung cd (a, p) = 1- Theo dinh ly Fermat thi: iVrim-B Chifng minh rang neu n > I la so le va p la so nguyen to le thi n oF-^ ~ 1 (mod p) ^ a'^(P-i) = 1 (mod p), \/k ^ 1. la luy thCca cua p. Chtfng minh. N e u a chia het cho p thi theo dieu k i e n bai toan suy Ta luon cd ra b chia het cho p. K h i do chia a, b cho luy thuTa cao nhat cua p , ,, n - Spin) ^ n - 1 trong chiing, ta l a i difa ve bai toan tren. Do do c6 the gia siJ a, b n - 1 khong chia het cho p. V i n le nen: Mat khac: (a + 6)|(a" + 5"). /n-l \l A:(p-l)^r! Gia suf ton tai u:dc nguyen to (? cua a + b. K h i do, q phai le, v i neu a — 1) = ^ ^ > ' = - i ) ^ y ; y^[a^ip-^) -1] q Chan thi (a" + 6 " ) : 2, trong khi do p'' f 2. K h i do theo dinh ly 2 'iJt thi: n n - l ^ Vpin^). v.ip'') = ^ , ( a " + b") = v,{a + b) + v,{n). L P - I J Yiq\{a + b) nen Vq{a + 5) ^ 1, din den w,(p'=) ^ 1, tiJc g | hay Tu" do ta cd dieu phai chiJng minh. q = p. Do do: • a + b = p"\ B a i tap 6 ( I M O , 1990). Xdc dinh tat cd cdc so tu nhien n > l t sao V i n ^ 3 nen a" + 6" > a + b, do 66 k > m va ta du'dc quan he: cho 2" + 1 chia het cho n^. k = m + Vp{n) => Vp{n) > 0. Lofi giai. G o i p la irdc nguyen to nhd nhat cua n . K h i dd (2" + 1 ) : p Gia sur Vp{n) = a > 0, tiJc n = p"/?, (/?, = 1. K h i do: nen p 16, dan den: • P = vpip") = vp{a- + 6") = vpia^"^ + = Vp{a''" + r). 2^" = 1 (mod p). Tii do suy ra: Theo dinh ly Fermat thi: a^^+lf ^p'^a'^ + b''. M a t khac. ta l a i c6 2"-^ = 1 (mod p). TiJf day, ta cd Suy ra /3 = 1, do do n = p^p("), dieu phai chufng tninh. • 2(2n,p-l)_i (inodp).
  20. 32 Cdc phuang phdp gidi todn qua cdc ky thi Olympic ndng luy thita vd ling dung 33 V i p n h o nhatnen (n, = 1, d i n den (2n, p - 1 ) = (2, = 2, Vay Vp{b) < Vp{c) va nhu the ta c6 ^/J?' turc la 2^ = 1 ( m o d p ) , suy ra p = 3. Dat n = S'^.k (k le, {k, 3) - 1). K h i do: Vp(c) = ^^p(2c + b)+ Vp{3c + b)^ Vp{b) + Vp{b) ^ Vp{c) ^ 2 • Vp{b). t;3(2" + 1) ^ V3{n^) v:i{2 + 1) + vsin) ^ 2a 2i^p(6) ^ Vp{c) nen ta suy ra Vp{c) = 2 • Vp{b). D i e u nay chiJng • , , • ^ 1+ a ^ 2Q; td m p i luy thijfa nguyen to cua p trong c deu 1^ so chan, tiJc c la so => Q = 1 chlnh phifcfng. • ' n = 3k. B a i tap 8. Cho a vd b Id cdc so nguyen ducfng sao cho: N e u A; > 1, g o i g la tfdc nguyen to nho nhat cua A; (g ^ 5). Suy ra: a\b\ 1 2^fc = - 1 ( m o d g) ^ 2^^^ = 1 (mod q) 2(^*--'''-i^ = 1 ( m o d q) Chiing minh rang a = b. W { k , q - l ) = l nen (6fc, g - 1) = 1 (loai) hoac (6fc, g - 1) = 2 ' hoac (6A;, g - 1) = 3 (loai) hoac (6/c, g - 1) = 6. V d i {6k, g - 1) = 2 Chufng minh. TO dieu kien bai toan suy ra p la \idc nguyen to' cua thi: a k h i va chi k h i p la ifdc nguyen to cua b. Ta chufng minh: 2^ = 1 ( m o d g) g = 3 (loai). Vp{a) = Vp{b). V d i (6/c, g - 1) = 6, ta c6 TO gia thiet thi a^^^+i I 6"^+^ I ^4fc+4 ^5 ^.5 2^ = 1 (mod g) g = 3 (loai) hoac g 7. i } Dan den 8'^ = - 1 ( m o d 7) (v6 l y ) . V a y = 1 , tiJc n = 3 la gia tri Vp{a''^') ^ Vp{b''+') => {4k + l)vp{a) < {4k + 2)vp{b) ' duy nhat thoa man yeu cau. • _^ Vp{a) 4k+ 2 B a i tSp 7. Cho a, b, c la ba so nguyen ducfng thoa man dieu kien: Vp{b) ^ 4k+l c{ac+lf = {2c + h){Zc + h). Chiing minh rang c la so chinh phucfng. Vp{b''+') ^ vp{a"'+') ^ {4k + 3)vp{b) ^ {4k + 4)vp{a) ^, , Vp{a) ^ 4fc + 3 Chiyng minh. TO dieu k i e n bai toan t h i : ^ ^ ^ 4 k T 4 - c{ac + 1 ) 2 = + 56c + b'^ =>b^':c. day suy ra: 4fc + 3 ^ Vp{a) ^ 4k+ 2 Ggi p la ifdc nguyen to tiiy y cua c thi ta c6 4A: + 4 ^ Vp{b) ^ 4fc + 1 ' ' ' ' 6^ ; c ^ Vp{b^) ^ Vpic) ^ 2 • Vp{b) ^ Vpic). Cho A; oo, ta difcJc Vp{a) = Vp{b). TO do suy ra a = 6. • Gia suf Vp{b) ^ Vp{c). K h i 66 do (ac + 1 , c) = 1 nen i;p(ac + 1) = 0. B a i tap 9. Chiing minh rang vdi moi so nguyen ducmg n, ta deu c6 Do d6: n-l y Up(c) = t;p(2c + 6 ) + t ; p ( 3 c + 6) ^ ' y p ( c ) + i ; p ( c ) n!| [](2"-2'=). ^ Vp{c) ^ 2 • vp{c) (v6 l y ) .
ADSENSE

CÓ THỂ BẠN MUỐN DOWNLOAD

 

Đồng bộ tài khoản
4=>1