Tài liệu ôn thi olympic Toán đại số

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Tài liệu ôn thi olympic Toán đại số được chia làm 6 chương, gồm những nội dung như: Đa thức, ma trận, định thức, hệ phương trình tuyến tính, không gian vecto, ánh xạ tuyến tính và tổ hợp. Mời các bạn cùng tham khảo!

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biên) – – – THANH PHONG
thi . sinh viên . T b [6], [9], [10]. Oly . 1
Ụ Ụ 1 Ụ Ụ 2 hươn 1 Ứ 6 1 Ứ .............................................................................................................6 .................................................................................................6 .....................................................................................................6 ..........................................................................6 .....................................................................................................7 .....................................................................7 ........................................................................................9 §2 Ủ Ứ ...............................................................................11 ..........................................................................11 .............................................................................................12 .........................................................................14 4. ................................................16 5. ..........................................................17 6 .........................................................................17 §3 Ứ ......................................21 1. ....................................21 2. ......................................................23 hươn 29 1 Ủ N................29 .........................................................................................29 .........................................................29 .........................................................................................29 ...............................................................................................29 5. ........................................................................................30 ......................................................................31 2
................................................................................................ 31 2. Ma .......................................................................................................31 .......................................................................31 .................................................................................................31 ..................................................................................................31 6. Ma tr ...................................................................................................32 ...................................................................................................32 ................................................................................................ 33 § ...............................................................................35 1. ............................................................35 ..........................................................36 Ừ ...................................................................................42 ................................................................................42 ........................................................42 NG Ủ ....................................................................................49 ................................................................................................ 49 ............................................49 3. ng c a ma tr n .............................................................50 hươn ......................................................................................................55 hươn Ứ 61 1 Ứ ........................................................................61 ...........................................................................................61 .................................................................................61 Ứ ............................................................64 .............................................................................64 ....................................................................................66 ..........................................................................................67 ............................................................70 ...............................................72 .............................................................................................73 3
hươn Ì 81 §1 Ì .....................................81 1. H n tính và không tuy n tính ................................................81 2. D ng ma tr n c a h n tính ....................................................81 3. Nghi m c a h ..............................................................................82 4. H n ........................................................................83 § Ì ....................................83 .............................................................83 ử Gauss .....................................................................................86 .........................................................................................89 4. Sử d nh lý v nghi m c c. ......................................................91 5. Sử d ix gi i h i x ng. ................94 .............................................................................................96 hươn Ô E TUY N TÍNH 108 §1 Ô E ..................................................................................108 1. Khái ni .............................................................................108 c l p tuy n tính và ph n tính ......................................................108 và s chi u c ...........................................................108 4. Ma tr n chuy t x1, x2 ,..., xn  sang  y1, y2 ,..., yn  ..............................109 5. Không gian con - H ng c a m t h ........................................................110 6. T ng và t ng tr c ti p.......................................................................................110 7. .............................................................111 §2. ÁNH X TUY N TÍNH ..................................................................................116 1. Khái ni m ánh x tuy n tính .............................................................................116 2. Ma tr n c a ánh x tuy n tính...........................................................................117 3. Ảnh và h t nhân c ng c u tuy n tính. ........................................................118 4. Giá tr ...................................................................................118 5. T ng c c ..............................................................................119 4
........................................................119 §3. CHÉO HÓA MA TR N VÀ ỨNG DỤNG ....................................................124 1. Chéo hóa ma tr n ..............................................................................................124 2. Ứng d ng c a chéo hóa ma tr n .......................................................................126 ng. .......................................................................................128 § ỨC CỰC TIỂU ....................................................................................134 c c c ti u ................................................................................................134 n c c ti u ....................................................................134 3. Bài t p áp d ng .................................................................................................135 ...........................................................................................136 hươn 6 TỔ HỢP 144 §1. CHỈNH HỢP – TỔ HỢP – HOÁN V ............................................................144 1. Ch nh h p..........................................................................................................144 2. T h p ...............................................................................................................144 3. Hoán v ..............................................................................................................145 §2. NH THỨC NEWTON – TAM GIÁC PASCAL ..........................................146 1. Nh th c Newton ...............................................................................................146 2. Tam giác Pascal ................................................................................................147 ỨNG MINH VÀ NGUYÊN LÝ QUY N P ..............148 ng minh tr c ti p và ph n ch ng ..........................................148 2. Nguyên lý qui n p .............................................................................................149 §4. NGUYÊN LÍ DIRICHLET - NGUYÊN LÍ CỰC H N ...............................152 1. Nguyên lí Dirichlet (hay còn g i là nguyên lí chu ng thỏ) .............................152 2. Nguyên lý c c h n ............................................................................................153 6 ................................................. Error! Bookmark not defined. 166 5
hươn 1 Ứ ,… 1 Ứ ửK K , . 1 h n hứ nh n h 1 K f ( x)  a0  a1 x  ...  an x n , ai  K , i  0,1,..., n a0 do. K K[x]. ủ hứ nh n h f ( x)  a0  a1 x  ...  an x n K n an  0 f (x n deg( f ử an f (x). h n ừ nh n h hứ n m nh n h f ( x)   ai x i ; g ( x)   bi x i i 0 i 0 n  m vaø ai  bi , i  0,..., n . n m nh n h f ( x)   ai x i ; g ( x)   bi x i i 0 i 0 6
max( m , n ) f ( x)  g ( x)   a i 0 i  bi  x i mn f ( x).g ( x)   ck x k , ck   ab i j k 0 i  j k nh 1 0  f ( x), g ( x)  K [ x] a) deg( f )  deg( g ) f ( x)  g ( x)  0 deg( f  g )  max{deg( f ),deg( g )}. deg( f )  deg( g ) f ( x)  g ( x)  0 deg( f  g )  d eg( f )  deg( g ). b) deg(fg)= deg(f )+deg(g). h h ư nh  K[x], g(x)  K[x] sao cho f ( x )  g( x )q( x )  r( x ), deg(r)  deg( g)  0. q(x), r(x f (x) cho g(x). nh n h f (x), g (x)  K [x], K g (x)  q(x)  K [x] sao cho f (x) = q (x)g (x f (x g (x) hay g (x f (x) trong K [x f (x) | g (x) hay g ( x) f ( x). h ng n nh ủ h hứ nh n h 6 ử 0  f ( x), g ( x)  K [ x] f (x g(x UCLN ( f ( x), g ( x )) h(x ỏ a) h (x h (x b) h (x) f (x g(x h (x) | f (x h (x) | g (x). c) f (x g (x h (x). nh n h f (x) g(x ▪ h nE ủ h hứ nh f ( x), g ( x)  K [ x] vaø deg( f )  deg( g ) a) = UCLN ( f ( x), g ( x))  b1g ( x), 7
b) r ( x)  0 UCLN ( f ( x), g ( x))  UCLN ( g ( x ), r ( x)) . . a) r (x) = f ( x)  g ( x) q ( x). UCLN ( f ( x), g ( x))  b1g ( x), b g(x). b) r ( x)  0 f ( x)  g ( x) q ( x)  r ( x). G ử h( x)  UCLN ( f ( x), g ( x)), h '( x)  UCLN ( g ( x), r ( x)) . h( x ) | f ( x) h( x) | g ( x) nên h( x ) | r ( x) h(x g (x r (x). Suy ra h( x) | h '( x) ’(x f (x g (x h '( x ) | h( x) h(x ’(x h( x)  h '( x). ■ nh f ( x), g ( x) u(x v(x) sao cho f ( x)u ( x)  g ( x)v( x)  1. hứn nh. ử f ( x), g ( x) , UCLN ( f ( x), g ( x))  1. ử deg( f )  deg( g ) n = deg(g UCLN ( f ( x), g ( x ))  1 u ( x ), v ( x ) sao cho f ( x )u ( x )  g ( x ) v ( x )  1 n = 0 hay g(x) = b0 u(x) = v( x)  b01 ỏ f ( x) u ( x)  g ( x) v( x)  1 . ử ỏ n, n > 0. f (x), g (x deg( f )  deg( g ) deg(g) = n q (x r (x) sao cho f ( x)  g ( x)q( x)  r ( x),deg(r )  deg( g ) neáu r ( x)  0. r ( x )  0 thì g(x r ( x )  0 , suy ra 1  UCLN ( f ( x), g ( x)) UCLN ( g ( x), r ( x)) deg(r )  deg( g )  n ’(x), ’(x) sao cho g ( x)v '( x)  r ( x)u '( x)  1 hay f ( x)u '( x)  g ( x)(v '( x)  q( x)u '( x))  1 u ( x)  u '( x); v( x)  v '( x)  q ( x)u '( x) f ( x)u ( x)  g ( x)v( x)  1. ử u(x), v(x) sao cho f ( x)u ( x)  g ( x)v( x)  1 UCLN ( f ( x ), g ( x )) f (x g (x UCLN ( f ( x), g ( x))  1. ■ 8
6. ụ hứ ụ 1. x 2017 cho ( x  2)3 trên [ x]. ử x 2017  ( x  2)3 q( x)  ax 2  bx  c (*). Thay x 4a  2b  c  22017. 2017 x 2016  3( x  2) 2 q( x)  ( x  2)3 q '( x)  2ax  b (**). Thay x = 2 (**) 4a  b  2017.22016 . x=2 a  2017.2016.2 2014 . b  2017.2015.22016 ; c  (1  1007.2017)22017. 2017.2016.22014 x2  2017.2015.22016 x  (1  1007.2017)22017. ụ 2. UCLN ( x m  1, x n  1)  x d 1 , d  UCLN (m, n); m, n  * d  UCLN (m, n) m ', n ' m  dm ', n  dn '. x m  1   x d   1; x n  1   x d   1 . Suy ra x m  1 x n  1 m' n' xd  1 xd  1 xm  1 xn  1. d  UCLN (m, n) u, v  sao cho um  vn  d ử h( x ) x m  1, x n  1 h( x) | ( x mu  1)  ( x nv  1) hay h( x) | x mu  x nv  x nv ( xd 1) UCLN ( xm  1, xnv )  1 nên UCLN (h( x), xnv )  1 h( x) | ( x d  1) . ụ f ( x)  [ x] m, n  * a) f ( xn ) x 1 f ( xn ) x n  1. b) a * ỏ f ( xn ) ( x  1)m f ( xn ) cho ( xn  an )m . 9
c) ử f (x x2  x  1 f ( x), h( x)  [ x] ỏ f ( x)  g ( x3 )  xh( x3 ). g ( x) x 1 h(x x  1. a) f ( xn ) x 1 f (1n )  f (1)  0 f (x) hay f (x x  1, f ( x)  ( x  1) g ( x) f ( xn )  ( xn  1) g ( xn ) f ( xn ) x n  1. b) f ( xn ) ( x  1)m f '( x n ) ( x  1)m1 f ( m1) ( x n ) ( x  1). f (an )  f '(an )  ...  f ( m1) (a n )  0. an f ( x), f '( x),..., f ( m1) ( x) f (x x  a n m , thay x xn f ( xn ) ( x n  a n )m . 1 3 c) , 2  i , 2 2 2 x 2  x  1. f (x x2  x  1 nên f ()  f (2 )  0. g (1)  h(1)  0 g (1)  2h(1)  0 g(1) = h(1) = g ( x) h(x x  1. . x100  2 x 51  1 cho x  1. 2 1. ư n n 1 §1, 6) - 2x + 2. 2. (USAMO 1976) Cho f (x), g (x), h (x), s (x f ( x5 )  xg ( x5 )  x2h( x5 )  ( x4  x3  x2  x  1)s( x) f (x x - 1. ư n n Thay x f (1) = 0. 10
3. ( [3] f ( x), g ( x)  [ x] ỏ f ( x 2010  2009)  x g ( x 2010  2009) x 2  x  1. f ( x), g ( x) x  2010. ư n n §1, 6). 4. x100  1 x 45  1 trên [ x]. x5 – 1. §2 Ủ Ứ 1 h hứ nh nh n h 1 ửK cK f ( x)  a0  a1 x  ...  an x n  K [ x]. ửc f ( x) f (c)  a0  a1c  ...  anc n  0 . f (x) trong K a0  a1 x  ...  an x n  0 trong K. nh 1 c  K , f ( x)  K[ x]. f ( x) cho x  c f (c). Khi chia f ( x) cho x  c ử K f ( x )  ( x  c)q( x )  r, q( x )  K[ x] Thay x = c f (x) f (c )  r . f ( x) cho x  c f (c). ■ 1. f ( x) f ( x) x  c. a, b  K , f (a )  f (b) a  b. 11
nh 2. . a không • ơ n Cho f ( x )  a0  a1 x  ...  an x  K [ x ]. n f ( x) cho x  c q( x )  b0  b1 x  ...  bn1 x n1 r  f (c). q ( x) r an an 1 ... a1 a0 c bn 1  an bn 2  an 1  cbn 1 b0  a1  cb1 r  a0  cb0 2 h hứ nh n h ử c m * f ( x)  K [ x] f ( x)  ( x  c)m g ( x), g ( x)  K[ x] g(c)0. nh 3. Cho f ( x)  K[ x] f (x).■ 2. ử f(x x ỏ n trên K n+1 K. Suy ra f (x) – g (x n f (x) - g(x n n f (x) – g(x f (x) = g(x).■ nh 4. x1 , x2 ,..., xk f ( x)  an xn  axn1  ...  a0  [ x] m1 , m2 ,..., mk m1  m2  ...  mk  n f ( x)  an ( x  x1 ) m ( x  x2 ) m ...( x  xk ) m . 1 2 k 12
f(x ( x  x1 )m , ( x  x2 ) m , ..., ( x  xk ) m . 1 2 k ( x  x1 )m , ( x  x2 ) m , ..., ( x  xk ) m 1 2 k f(x ( x  x1 )m ( x  x2 ) m ...( x  xk ) m . 1 2 k f ( x)  q( x)( x  x1 )m ( x  x2 )m ...( x  xk ) m , 1 2 k so f (x .■ nh 5. ) f ( x)  an x n  an1 x n1  ...  a1 x  a0  K [ x], an  0 x1 , x2 ,..., xn  K an 1 x1  x2  ...  xn  an an  2 x1 x2  x1 x3  ...  xn 1 xn  an ... (1) n a0 x1 x2 ... xn  an ử f(x n x1 , x2 ,..., xn , f ( x)  an ( x  x1 )( x  x2 )...( x  xn )  an x n  an ( x1  x2  ...  xn ) x n1  ...  (1)n an .x1x2 ...xn .  an 1  x1  x2  ...  xn  a  n  an 2  x1 x2  x1 x3  ...  xn 1 xn  a  n ...   (1) n a0 x x  1 2 n... x   an ■ 13
3. hứ h n ư n nh n h 1 ử K f (x)  K [x trong K [x K [x f (x) = g ( x )h ( x ), g (x), h (x)  K [x g (x) hay h (x nh 1. . f (x g(x R[x] sao cho f ( x) g ( x)  1 R 0  deg( fg )  deg( f )  deg( g ) deg( f )  deg( g )  0 , hay f ( x) f ( x) f (x ử R .■ ụ x+ [ x] [ x] . nh 6. a) b) a) f ( x)  ax  b  K [ x], a  0 . f ( x)  0 f ( x) ử f ( x )  g ( x ) h( x ) g ( x), h( x)  K [ x] , deg(g) + deg(h) = deg( g ),deg( h)  deg( f ). deg(g) = 1; deg(h) = deg(g) = 0; deg(h) = 1. deg(g) = 1; deg(h) = h(x deg(g) = 0; deg(h) = g(x f ( x) f(x .■ ử f (x K f (x  f (x)= g(x)h(x); g(x), h(x)K[x g(x h(x  f (x) K. f(x) f(x) K. ■ 14
nh  x , . 6 2,  x ử f (x  x 2 2, 1), f (x f ( x )  ( x  c)g( x ),deg(g)  0, g( x )  [ x], x–c f (x f (x .■ nh f ( x)  [ x] z  a  ib, b  0 z  a  ib f ( x) x 2  2ax  a 2  b 2 . ử f ( x)  an xn  an1xn1  ...  a0 z f ( x) f ( z)  an z n  an1z n1  ...  a1z  a0  0.   n n 1 an z  an1 z  ...  a1 z  a0  0 z f (x f (x ( x  z)( x  z)  x2  2ax  a 2  b2 , a, b  . ■ [ x], 6 2, 3  x . ử f (x  x f (x c f ( x)  ( x  c) g ( x), deg(g)  0, g( x )  [ x]. xc f (x f (x f (x 2 2, 1), f (x z 2, 3) suy ra f (x g ( x)  ( x  z)( x  z)  x2  ( z  z) x  z  z, 15
g(x   0. Do f (x f ( x)  k g ( x), k  \ {0}. f (x   0. ■ 4. h h ủ hứ h h ỏ f ( x )  n x n  n1 x n1  ...  0 , n  0, i  , i  1, n. 1 1 f ( x )  (an x n  an1 x n1  ...  a0 )  g( x ), b b b ai  , i. f (x g (x  g(x ann  an1n1 ...  a0  0, an  0 hay  an   an1  an  n 1 ...  a0 ann1  0. n Suy ra an x n  an1 x n1 ...  a0 ann1  0 (*). f (x nh f ( x )  an x n  an1 x n1  ...  a0 , n  0 p  , UCLN( p, q)  1 0 q n. .  f (x n n 1  p  p  p  p f    an    an1    ...  a1    a0  0 q q q q Hay a0 q n   p(an p n1  an1 p n2 q  ...  a1q n1 ), v an p n  q(an1 p n1  ...  a1 pq n2  a0 q n1 ) . p a0 q n q a an p n UCLN (p, q) = 1 nên p a0 q an.. ■ 16
nh f ( x )  x n  an 1 x n 1  ...  a0 , n  0 a0  1  (1), 1   -1). p  , UCLN ( p, q)  1 f (x p q a0 q  a0.  f (x f ( x )  ( x  )g( x ) g( x )  [ x]. f (1)  (1  )g(1) f (1)  (1  )g(1) 1  f 1  f (-1). ■ ▪ n h h ủ hứ f ( x )  x n  an1 x n1 ...  a0 , ai  như f (1); f -  f (x) không? f (1) f (1)  a0 ỏ ; 1  1  f (x) không? 5. hứ h n ư n h [x Eisenste h n Eisenstein. f ( x )  an x n  an1 x n1 ...  a0 , an  0 (n  1) n 2 0 quy trong [x]. 6 ụ n h hứ ụ 1. Cho m f (x) = x5 – x + m [ x]. . 17
ử f ( x)  g ( x)h( x), g ( x), h( x)  [ x]. m 5 a a5  a  0(mod5) f (x deg( g ), deg(h)  2. g, h .G ử  x1  x2  b  g ( x)  x  bx  c  [ x] c  x1 x2  c 2 x 1 , x2  x 5  x  m  0, i  1, 2  i i Suy ra x15  x25  x1  x2  2m  0 x1  x2 ; x15  x25 .  x1  x2   x1  x2 (mod5) . 5 ( x1  x2 )5  x15  x25 (mod 5) suy ra x15  x25  x1  x2 (mod5) hay m f (x [ x]. ụ 2. n f ( x)  ( x  1) 2 n 1 x n2 c x  x  1. 2 1 3 x2  x  1    i . x2  x  1 f (x 2 2 x2  x  1 f ( )  0 . V n 2 n 1 n2 1 3  1 3 f ( )    i   i  2 2   2 2   (2n  1)   (2n  1)    ( n  2)2   ( n  2)2   cos    i sin    cos    i sin  0  3   3   3   3   (2n  1) (n  2)2   do     3 3  f ( x)  ( x  1)2 n1  xn2 c x2  x  1. ụ 3. ( 2012, [5]) Cho m, n xm  xn  1 x2  x  1 mn  2 cho 3. 18