1. 8o 2014 13 - 25 2014 OLYMPIAN BAY -

2. 2014. . . , . , , . . 2014

3. . , . 1 - 44 . , . .45-68 III. , . ... 69-82 IV. , . . .83-90 V. , . ..91-122 VI. , . .....123-158 VII. , . ...159-236 VIII. , . .. 237-280 IX. : 281-286

4. . . , . ( ) ... 1 0 , , 0,1,..., n f x = an x + + a x + a ai ai ^ i = n , x . x (indeterminate), ( ) 0 1 ... ..., , 0,1, 2,... n f x = a + a x + + an x + ai i = = ^ _, . i a . ( ) 0 1 2 n 1. 2 f x = a + a x + a x + + anx + k , ak 0 an = 0 n > k. deg f (x) = k f (x) = k . f x = a + a x + + an x + ( ) 0 1 ... ... m 1 2. ( ) 0 1 ... ... n g x = b + b x + bmx + , (1) ai ,bi , 0, 0 0, 0, 1, 2,3,... n ni m m i a a b b i + + = = = : ( ) ( ) f x = g x n = m ai = bi i = 0,1, 2,..., n . x , [ ] { } 0 1 ... ... : , 0,1, 2,... n x = a + a x + + an x + ai i = , ( ) ( ) ( 0 0 ) ( 1 1) ... ( ) ... n f x + g x = a + b + a + b x + + an + bn x +

5. 2 2 f (x)g(x) = a0b0 + (a1b0 + a0b1)x + (a0b2 + a1b1 + a2b0 )x + ... , f (x), g ( x) (1) 2 . f (x) = a0 , a0 0 , 0. [x] f (x) = 1 . [x] 0(x) = 0 + 0x + 0x2 + ... 0(x) = 0. . f (x), g(x)[x] , : deg[ f (x) + g(x)] max{deg f (x), deg g(x)} deg f (x)g(x) = deg f (x) + deg g(x) . f (x) g(x) , deg 0(x) = . [x] , g (x),h(x)[x] : g (x)h( x) = 0( x). [x] . : 1. () g (x),h(x)[x] : g(x)h(x) =0(x). g (x) = 0(x) h( x) = 0( x) . () g(x),h(x), f (x)[x] f (x) 0(x) : g(x) f (x) =h(x) f (x) , g(x) =h(x) ( ). () g (x) = 0(x) , . g (x) 0(x) , h(x) = 0(x). , h(x) 0( x), g(x)h(x) =0(x) deg g (x)h(x) 0 , .

6. () g (x) f (x) =h(x) f (x)(g(x) h(x)) f (x) = 0(x),, f (x) 0(x) , () : g (x) = h( x) . , : 2. ( ). f (x), g(x)[x] g(x) 0 , p(x) (), (x) ()[x] , f (x) = g(x) p(x) + (x), = + + + + 1 = ( ) ( ) ( ) ( ) 1 1 x = f x g x p x . x g x p x x , deg x deg x = + 1 2 2 2 1 ................................................................................ x g x p x x x x x , deg deg 0. k k k k k = + < = + + + + 3 (x) = 0 deg (x) < deg g(x) . ( (x) = 0 , g(x) f (x) : g(x) f (x) .) deg f (x) < deg g (x) , p(x) = 0( x) (x) = g (x) . . deg f (x) = n m = deg g (x) , f (x) g ( x) . ( ) n n 1 ... 1 1 0 f x anx an x a x a ( ) 1 ... 1 0 m m g x bmx bm x b x b = + + + + , f ( x) g ( x) p ( x ) a x 1 n n m m b ( ) 1 x = 0 ( ) ( ) 1 deg x < deg g x , ( ) ( ) 1 p x = p x ( ) ( ) 1 x = x . ( ) ( ) 1 deg x deg g x , ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 f x = g x p x + x , deg x deg g x ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 f (x) = g (x) p(x) + (x), (x) = 0 deg (x) < deg g (x)

7. p(x) = p1 (x) + p2 (x) +...+ pk (x) ( ) ( ) k 1 x x + = . . p(x) (x) f (x) = g (x) p( x) +(x), (x) = 0 deg(x) < deg g (x) = m. + = + = x + 3 x + x 2 x 1 x 1 x x x x + + + 4 ( ) ( ) ( ) ( ) ( ) ( ) ( )( ( ) ( )) ( ) ( ) (*) g x p x x g x p x x g x p x p x x x p(x) p( x) 0, p(x) p( x) , deg g ( x)( p ( x) p( x)) m deg ( x) ( x) < m, (*). p(x) = p(x) , (*) (x) =(x) . , x4 + 3x3 + x2 2x 1 = (x2 1)(x2 + 3x + 2) + ( x +1) , : 4 3 2 2 4 2 2 3 2 3 2 2 3 2 ( ) 3 2 2 1 3 3 2 1 2 2 1( ) x x x x x x x x x + + + + + + + f (x)[x], x . x = f ( ) = 0 , f (x) . x f (x) (x ) f (x) : f ( x) = ( x ) g ( x) , g ( x)[x].

8. 1. () f (x) g ( x) = x a = f (a) . () f (x) g ( x) = ax + b, a 0, f b = 5 a . , f (x) , f (x)^[x] , n , ^ . , f (x) n ^ . f (x) = 0 f (x) . , ( ) ... 1 0 , , 0,1,..., n f x = an x + + a x + a ai ai ^ i = n , ( ) ( )( ) ( ) 1 2 , n n f x = a x x x 1 2 , ,..., , n ^ . : ( ) ( ) ( ) 1 2 , ,..., , n x x x f (x) ( )( ) ( ) 1 2 n x x x f (x) . 3. ^ f (x)[x] k , : ( )k ( ) x f x ( ) 1 | k x + f ( x) . : 3. ^ k f (x)[x], , , ( x)[x] : ( ) ( )k ( ) f x = x x ( ) 0 . ( ) ... 1 0 , , 0,1,..., n f x = an x + + a x + a ai ai ^ i = n 1 2 , ,..., , k ^ k n , 1 2 , ,..., , k , :

9. ( ) ( ) 1 ( ) 2 ( ) 1 2 k , n k f x a x x x = 1 2 ... . k + + + = n f (x) = 0 , f (x)[x], , . ^ , , , , x2 +1 = 0 . : 4. f ( x)[x] deg f = n n +1 , f ( x) = 0, f (x) . 5. f (x), g(x)[x] , n , n + 1 x , , : ( ) ( ), 1, 2,..., , 1 i i f = g i = m m n + 1 2 , ,..., m , f (x) g(x) . ( ) ( )2 ( ) ( )2 ( ) ( )2 ( ) ( )( )( )f x = x + x + x + ,,, , , x , , , . , ( ) ( )2 ( ) ( )2 ( ) ( )( )( ) f = + + =0, f ( ) = f ( ) = 0 . Vieta 1 2 , ,..., n n ( ) 1 0 n .... , , 0,1, 2,..., , 0, n i n f x = a x + + a x + a a ^ i = n a 6 ( Vieta):

10. f x = a x + + ax + a a x a x a x a a = + + + + a a a f x a x x x = n n = x n n x + x n + n = . 2 1 ... ... i j n n 2 1 ... 1 ... 2 1 ... 2 1 2 2 , 2 7 ( ) 1 1 1 2 2 2 1 2 1 3 1 1 0 1 2 ... ... ... ....................................................................... ... 1 n n n n n n n n n n n a a a a a a = + + + = = + + + + + = = = ( ) 1 0 1 1 1 0 .... , 0, n n n n n n n n n n ( ) ( )( ) ( ) ( ) 1 2 1 2 1 2 1 . n n . ( ) 1 2 2 3 3 1 0 n 2 n 2 n n ... n f x x nx n x a x a x a = + + + + + + . f ( x) . . 1 2 , ,..., n f ( x) . , Vieta, : 1 2 ... 2 n + + + = n 2 1 2 i j i j n n < ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 1 2 1 2 1 2 2 1 2 1 2 2 2 1 2 i j n n n n n n n n n n n n < = = + + + + + + + + + + + + = + + + = = .

11. ,_,] . , . . 6. ( ) 1 0 n .... , , 0,1, 2,..., , 0, n i n f x = a x + + a x + a a i = n a , = a + bi,b 0, = a bi. f a bi a a bi a a bi a a j n a f a bi a a bi a a bi a .... 0, , 0,1,2,..., , 0, ... + = + + + + + = = + = + + + + + a a aj n f a bi a a bi a a bi a f a bi a a bi a a bi a ( , 0,1,2,..., ) ... 0, ... 0 + = + + + + + = = + + + = 8 . , ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 0 1 0 1 0 1 0 n n j n n n = = j j j n n n n = a bi f ( x) . , 6 : 1. f ( x ) = a x n + .... + a x + a , a , i = 0,1,2,..., n , n 1 0 i 0, n a , k n = a + bi,b 0, , k = a bi. 2. . 3. . 2 3 . 1 :

12. ( ) 1 0 n .... , 0, n n f x = a x + + a x + a a , k n = a + bi,b 0, a bi, ( k f ( x) ) : ( ) ( )( ) ( ) 1 f x = x x g x ( ) 1 g = 0 . : ( ) ( )( ) ( ) 1 2 g x = x x g x ( ) 2 g = 0 ( ) ( )( ) ( ) 2 3 g x = x x g x ( ) 3 g = 0 .................................................................... ( ) ( )( ) ( ) k 1 k g x x x g x = ( ) 0 k g = ( ) ( )( ) ( ) k k 1 g x x x g x + = ( ) 1 0 k g + . ( ) 1 0 k g + , , ( ) 1 0 k g + = , ( ) k 1 g x + , , ( ) 1 0, k g + = . , . . 7. f ( x ) = a x n + .... + a x + a , a _ , i = 0,1,2,..., n , n 1 0 i 0, n a , f ( x)_[x], x = a + b,b > 0, b _ , a b. f ( x) ( ) ( )( ) ( )2 x = xa b xa+ b = xa b, f ( x) = ( x) ( x) + x + , (1) ( x), x + _[x]. x = a + b (1) : f (a + b ) = (a + b )+ = 0( a + ) + b = 0 a + = 0, = 0 = = 0, (1) f (x) = (x) (x) = (x a b )(x a + b ) ( x) . (2) (2) a b f ( x) . , 9

13. 6 10 7. , , . 1. ( ) 1 0 n .... , , 0,1,2,..., , n i f x = a x + + a x + a a _ i = n 0, n a , k n z = a + b,b > 0, b _ , k z = a b. 2. . 3. . , . 4. f (x)[x] , g (x) h(x) , 1, f (x) = g (x)h(x) . 5. f (x) = 0 , f (x)[x], deg f (x) 1 , . ^ , , , , x2 + 1 = 0 . : 8. ( = ^ _). [x] : () ax + b, a 0 . () ax + b, a 0 [x], , , , , , .

14. () ax + b = g (x)h(x),a 0, deg g ( x) 1, deg h(x) 1, 2, 1, . ax + b, a 0 . () [x] . f (x)[x] ax + b, a 0 , f (x) = 0 x b 11 = , a . , , . p(x) . p(x) = ( x )q(x), q(x)[x]. p(x) , q(x) = c, c 0 , ( ), p(x) = ( x )c = cx c , . , ^ ( ), . [x] , 5, x x2 + bx + c b2 4c < 0 . _[x] . , . f (x)_[x]. f ( x) , , f (x)][x], . , . Gauss . f (x)][x] (primitive), . f (x)][x] g (x)][x] . .

15. Gauss. f (x)][x] _, ] . ( ) ( ) ( ) [ ] 1 0 n .... , 0, n n f x = a x + + a x + a = g x h x ] x a g ( x) h(x) . b c g (x) h(x) , , bg ( x) ch(x) bcf (x) =bg(x)ch(x) bcf (x) ] . f (x) ] . ( ) 1 0 k ... k bg x = b x + + b x + b ( ) 1 0 m ... 12 m ch x = c x + + c x + c . p b . f (x) p . i 0 1 1 , ,..., , i pb pb pb p | i b . ( ) 0 0 0 : ... mod i i i i b a =b c + + b c b c p , 0 p c . ( ) 1 0 1 1 10 1 : ... mod i i i i i p a b c bc b c bc p + + + = + + + , 1 p c . j p c , j . , c h( x) p , bc f (x) ] . p p f ( x) ] . , 9. ( ) 1 0 n .... , , 0,1, 2,..., , n i f x = a x + + a x + a a ] i = n 0, n a , f (x)][x], 0 , ]* , `* ,( , ) = 1, 0 a n a . , ( ) 1 0 n .... , , 0,1,2,..., , n i f x =a x + +a x+a a ] i = n 0 n a , , .

16. n n = + + + = f a a a a .... 0 ... 0. ... 0 mod , 1 1 0 n n n 1 1 n n a a a a + + + + = 1 1 0 n n n a a a a = + + + = n + n 1 + + n 1 + n = f a a a a 0 ... 0. 1 1 0 n n n n n n n a a a a a a ... , , 1. = + + + = n n n n = + + + + = , g (x) 13 1 n n ( 1 1 ) ( ) n n 0 1 1 . : ( 1 ) ( ) 1 1 0 , =1, ( ) f ( x). , . x4 4 = (x2 2)(x2 + 2) . _[x] . , . . 10. ( Eisenstein) ( ) 1 [ ] 1 1 0 n n .... . f x = a x + a x + + a x + a x n n ] p : (i) (i) p | , (ii) , 0,1, 2,..., 1 (iii) 2 | n i a p a i = n p 0 a , f ( x) _. f ( x) _, ]. ( ) 1 ( ) ( ) 1 1 0 n n .... n n f x a x a x a x a g x h x h(x) , ( ) ( ) ( ) 1 0 1 0 r ... , s ... , deg , , 0 r s g x = b x + +b x +b h x = c x + + c x + c r + s = n = f x r s > .

17. f (x) = g (x)h(x) : 0 0 0 0 11 0 , ... n r s i i i i a bc a bc a bc b c bc = = = + + + . p 0 a , p2 0 a , 0 0 0 a = b c 0 0 b ,c p , 0 p b p | 0 c . , n r s a = b c p | n a , p | r b p | s c . g (x) p p . i b p i , i > 0 . 0 11 0 ... i i i i a bc b c bc = + + + modulo p , ( ) 0 0 mod i b c p , , p | | ib p 0c . 1. f (x) = x3 4 _, x = y +1 f ( y) = ( y +1)3 4 = y3 + 3y2 + 3y 3 , , Eisenstein _, p = 3 . 2. p , f (x) =1+ x +...+ xp1 _ . , ( ) ( ) 1 2 1 1 + + 1 = = + + ... + + 1 2 14 p p p x p p fx x x x p x p . p f ( x +1) Eisenstein. { } = f1 (x), f2 (x),..., fm (x) [x] . (x) , : (i) (x) (ii) (x) , . (iii) (x) 1.

18. . (iii) . , (x) (i) (ii), c (x) , c . . 11. ( ) ( ) 1 2 x x ( ) ( ) 1 2 f x f x , , (x) 0 , : ( ) ( ) ( ) ( ) ( ) 1 2 1 2 x f x f x x = x : ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 2 2 2 f x =q x x + x f x =q x x + x ( ) ( ) ( ) ( ) ( ) ( ) 1 1 2 2 deg x < deg x x = 0, deg x < deg x x = 0. ( ) ( ) ( ) 1 2 x f x f x . ( ) ( ) 1 2 f x f x (x) . ( ) ( ) ( ( ) ( )) ( ) ( ) ( ) 1 2 1 2 1 2 f x f x = q x q x x + x x , ( ) ( ) { ( ) ( )} ( ) 1 2 1 2 deg x x max deg x ,deg x < deg x ( ) ( ) ( ) ( ) 1 2 1 2 x x = 0 x = x . , ( ) ( ) 1 2 x = x . ( ) ( ) ( ( ) ( )) ( ) ( ) ( ) ( ) 1 2 1 2 1 2 f x f x = q x q x x x f x f x . , . f (x) = q(x) (x) + (x) f (x) ( x) (x) . f ( x) g (x) . : 15

19. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) f x = q x g x + x x < g x 1 1 1 g x = q x x + x x < x 2 1 2 2 1 x = q x x + x x < x x = q x x 1 1 1 1 16 2 1 , deg deg , deg deg .............................................................................. , deg deg k k k k k k k k + + + + : ( f ( x) , g ( x) ) = ( ) k 1 x + . . 12. { } [ ] = f1 (x), f2 (x),..., fm (x) x , , (x) = g1(x) f1(x) + g2 (x) f2 (x) + ...+ gm (x) fm (x) , [ ] g1(x), g2 (x),..., gm (x) x . f (x), g(x)[x] (relatively prime), 1. 12 (x) f (x) + (x)g(x) = 1, (x), (x)[x]. 1. f ( x) ( ) 1 g x ( ) 2 g x , . . ( ( ) ( )) 1f x , g x =1, 12 a (x) b(x) : ( ) ( ) ( ) ( ) 1 f x a x + g x b x =1 . (1) (1) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 1 2 2 f x a x g x + g x g x b x = g x . (2) (x) f ( x) ( ) ( ) 1 2 g x g x , (2) ( ) ( ) 2 x g x .

20. ( ) ( ( ) ( )) 2 x f x , g x =1, deg (x) 1, . 2. f ( x) g (x)h( x) g ( x) , f (x) h(x) . . : h(x) = 0 . f (x) h(x) . h(x) 0. , ( f (x), g (x)) =1, a (x) b( x) f (x)a(x) + g ( x)b(x) =1, : f (x) a(x)h(x) + g (x)h(x) b(x) = h(x) . f ( x) , f (x) h(x) . 3. f ( x) g ( x) h(x) , h(x) . . f (x) h(x) , (x) h(x) = f (x) ( x) . g (x) h( x), g (x) f (x) ( x) . ( f (x), g (x)) =1, g (x) (x) . (x) (x) = g (x) ( x) . : h(x) = f (x) g (x) (x) , f (x) g (x) h(x). 11 . 12. ( ) ( ) ( ) 1 2 , ,..., n x x x ( ) ( ) ( ) 1 2 , ,..., n f x f x f x , , (x) . : () ( ) ( ) ( ) ( ) 1 2 ... : n f x + f x + + f x x ( ) ( ) ( ) ( ) 1 2 ... n x = x + x + + x . 17

21. () f1 ( x) f2 ( x)... fn ( x) : ( x) ( ) ( ) ( ) ( ) 1 2 ... : n x x x x . () ( ) : ( ) ( ) : ( ), 1,2,..., k k i i f x x x x = n f x f x ... f x x p x p x ... p x x x ... x + + + = + + + + + + + n n n 1 2 1 2 1 2 f x f x f x x p x x + + + = + 18 . () : ( ) ( ) ( ) ( ), 1,2,..., i i i f x = x p x + x i = n , (1) deg ( ) deg ( ) i x < x ( ) 0 i x = . (1) : ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 ... , n ( ) ( ) ( ) ( ) 1 2 ... n x = x + x + + x , ( ) ( ) ( ) ( ) 1 2 ... n p x = p x + p x + + p x ( ) ( ) ( ) ( ) ( ) 1 2 deg deg ... deg n x = x + x + + x < x , . () (1) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 1 2 ... ... n n f x f x f x = x x + x x x , (x)(x) (1) ( ) ( ) ( ) 1 2 ... n x x x . ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 1 2 ... ... , n n f x f x f x x x x = x x , 11, . () () ( ), i f x i =1, 2,..., n . , f (x) = x4k+3 + x4l+2 + x4m+1 + x4n , k,l,m,n , g (x) = x3 + x2 + x +1. x4 g (x) = x3 + x2 + x +1 1, , 12.(), ( 4 ) : ( ) k x g x 1k : g ( x)

22. . ( 4 ) 3 : ( ) k x x g x 1k x3 : g (x) , ( ) 1 x . ( 4 ) 2 : ( ) l x x g x 1l x2 : g (x) , ( ) 2 x , ( 4 ) : ( ) m x x g x 1m x : g ( x) ( ) 3 x , ( 4 ) : ( ) n x g x 1n : g (x) ( ) 4 x . , 12 (), ( ) 4 3 4 2 4 1 4 ( 4 ) 3 ( 4 ) 2 ( 4 ) ( 4 ) : ( ) f x = x k+ + x l+ + x m+ + x n = x k x + x l x + x m x + x n g x (1k x3 +1l x2 +1m x +1n ) = (x3 + x2 + x +1) = g (x) : g (x) ( ) ( ) ( ) ( ) ( ) 1 2 3 4 x = x + x + x + x . g (x) : g (x) 0, f (x) = x4k+3 + x4l+2 + x4m+1 + x4n : g (x) g (x) f (x) . 13. f (x)[x], = ^ ( ) 1 f x , ( ) ( ) 2 1 ,..., n f x f x ( ) ( ) ( ) ( ) ( ) ( ) 1 2 : , : ,..., : , n f x x a f x x a f x x a n n n f x x a px x a x a px f x x a px = = = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = = = 1 2 2 f x x a p x x a x a p x f x x a p x 1 2 ......................................................................................................................... f x x a p x x a x a p x f x x a p x = = = 19 . : ( ) ( ) ( ) ( ) ( ) 1 2 0, 0,..., 0 n n x a f x f a f a f a = = = . . ( ) ( ). n x a f x p(x) ( ) ( )n ( ) f x = x a p x . ( ) ( ) ( ) ( ) ( ) 1 ( ) ( ) ( ) 1 ( ) 1 ( ) ( ) 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 1 , n n n n n : ( ) ( ) ( ) 1 2 0, 0,..., 0 n f a f a f a = = = .

23. , ( ) ( ) ( ) 1 2 0, 0,..., 0 n f a f a f a = = = . ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) f x x a f x f x x a f x ................................. f x x a f x n n n + n n n f x nx n x nx x x ( 1) 1 1 1 1 ... 1 , = + + = = n n n 20 1 1 2 1 = = = : ( ) ( )n ( ) n f x = xa f x . ( ) ( ). n x a f x , n 1, f (x) = nxn+1 (n +1)xn +1 g ( x) = xn nx + n 1 ( ) ( )2 x = x 1 . f (1) = n (n +1) +1= 0, ( x 1) f ( x) . ( ) 1 ( ) ( ) ( x )( nx x 1 x 2 x ) ( ) ( ) ( ) 1 f x = x 1 f x , ( ) 1 2 1 f x = nxn xn xn ... x 1. ( ) 1 2 1 f 1 = n (1n +1n +...+1+1) = 0, : ( ) ( ) 1 x 1 f x ( ) ( )2 ( ) x = x 1 f x . g (1) = 0 g ( x) = xn nx + n 1 = (xn 1) n( x 1) = ( x 1)(xn1 + xn2 + ...+ x +1 n), ( ) 1 2 1 g x = xn + xn +...+ x +1 n ( ) 1 g 1 = 0 . f (x) xn a, n 1 ( ) 1 [ ] 1 1 0 k k ... , k k f x a x a x a x a x = + + + + = ^, k . 1 n k , f ( x)

24. ( ) 1 ( ) 2 ( ) ( ) ( ) f x = x n f x n + x n f x n + ... + xf x n + f x n , n 1 n 2 1 0 5 4 3 2 3 4 5 2 f x x x x x x x x x x x 3 2 5 6 2 6 3 5 1 3 5 2 6 . = + + + + = + + + + = + + + + = + + x x x x x xf x xf x f x = , ( ) i x 21 ( n ), 0,1,..., 1 i f x i = n xn . , f ( x) modulo n . , f (x) = x5 3x4 + 2x3 + x2 5x + 6 , n = 3, : ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 3 3 3 2 3 3 3 2 1 0 14. ( ) 1 ( ) 2 ( ) ( ) ( ) 1 2 1 0 n n n n ... n n . f x = x f x + x f x + + xf x + f x n n : () f ( x) xn a ( ) 1 ( ) 2 ( ) ( ) ( ) x = x n f a + x n f a + ... + xf a + f a . n 1 n 2 1 0 () ( ) ( ) ( ) ( ) ( ) ( ) 1 2 1 0 n ... 0 n n x a f x f a f a f a f a = = = = = . () 12 (), f (x) : (xn a) ( x ) 1 ( x ) 1 n i i = i ( n ): ( n ), 1, 2,..., 1 i x f x x a i = n . ( n ): ( n ) f x x a , i = 1, 2,..., n 1 f ( a ) , i i xi : (xn a) , i = 1, 2,..., n 1 xi . 12 () x i f ( x n ): ( x n a ), i = 1, 2,..., n 1 i i ( ) : ( n ), 1, 2,..., 1 i x f a x a i = n , i ( ), 1,2,..., 1 i x f a i = n , deg i ( ) deg ( n ) i x f a < x a . ( ) 1 ( ) 2 ( ) ( ) ( ) 1 2 1 0 n n ... . x = x f a + x f a + + xf a + f a n n () (). ,

25. . f (x) = x5 3x4 + 2x3 + x2 5x + 6 g (x) = x3 1. . f (x) ( ) 5 4 3 2 ( 3 ) ( 4 ) ( 5 2 ) f x x x x x x x x x x x 3 2 5 6 2 6 3 5 1 3 5 2 6 , = + + + + = + + + + = + + + + = + + ( ) ( ) ( ) ( ) ( ) ( ) 2 3 3 3 2 3 3 3 x x x x x xf x xf x f x = + + + + . 22 2 1 0 , 14 (), : ( ) 2 ( ) ( ) ( ) 2 2 1 0 x = x f 1 + xf 1 + f 1 = 2x 2x +8 . . . . ( ) 1 [ ] 1 1 0 n n .... , , (1) n n f x a x a x a x a x = + + + + = ^ ( ) ( ) 1 ( ) 2 1 2 1 n 1 n ... 2 (2) n n Df x f x na x n a x a x a D:[x][x], f (x) Df (x) = f (x) : 1. D( f (x) + g ( x)) = Df (x) + Dg (x) 2. D( f (x)) = Df (x), 3. D( f (x) g (x)) = Df (x) g (x) + f (x)Dg ( x) 4. D( x) =1 , Df (x) , . 1 4, (2). , [ ] ( ) ... 1 0 , n f x = an x + + a x + a ^ x

26. 1,2 ,...,k ^, k n , 1 2 , ,..., , k , : ( ) ( ) 1 ( ) 2 ( ) 1 2 k , n k f x a x x x = 1 2 ... . k + + + = n f ( x) . f (x)[x] . 15. f (x)[x] , ^ f ( x) , , , f ( ) = f ( ) = 0 f ( ) 0. , , ^ f ( x) , 2, , , : ( ) ( )2 ( ) ( ) f x = x x 0. (1) f ( x) . (1) , ( ) ( ) ( ) ( ) ( ) = + f x x x x x ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = + + + . f x x x x x x x x 23 2 2 2 2 2 2 . f ( ) = f ( ) = 0 f ( ) = 2 ( ) 0. , : f ( ) = f ( ) = 0 f ( ) = 2 ( ) 0. : ( ) ( )2 ( ) ( ) ( ) f x = x x + x , x = x + . (1) (1) x = : f ( ) = ( ) = + = 0 . (1) ( ) ( ) ( ) ( )2 ( ) f x = 2 x x + x p x + , (2) ( ) ( ) ( ) ( ) ( ) ( ) ( )2 ( ) f x = 2 x + 2 x x + 2 x x + x x (3)

27. x = f ( ) = = 0 f ( ) = 2 ( ) 0 , = f ( ) = 0, ( ) 0.. (1) : ( ) ( )2 ( ) ( ) f x = x x , 0 , (4) , , f ( ) = 0 , , f (x) f (x) . . f (x) Taylor, , . 16. ( Taylor ) f (x)[x], = ^,deg f (x) = n , , : ( ) ( ) ( )( ) ( )( ) f f f n f x f x x x = + + ++ . 24 ( n ) 2 ( ) ( ) n 1! 2! ! [ f ( x) f (x) f (n) (x) : ( f (n 1) (x)) = ]. Taylor : 17. = ^ k , , , f ( ) = f ( ) = ... = f (k1) ( ) = 0 f (k ) ( ) 0. Bolzano Rolle, .

28. 18 ( Bolzano). f (x) , < f ( ) f ( ) < 0 , ( , ) f ( ) = 0 . : 19. f (x)[x] , , < , (( ) , ) : ( ) = ( ) + ( ) ( ) + ( ) . n n f x f a a x a a x a a x a ( ) ( ) ... ( ) = + + + n n n n a x a a x a a x a a a a x a x a + + + n n + + + = 1 1 0 n ... n n n a x a x a a x 25 2 2 f x f f f . a , x . . 20. f (x)[x] a , > 0 f (x) f (a) x a , x x a 1, x[a 1, a +1]. ( ) 1 [ ] 1 1 0 n n ... n n f x a x a x a x a x = + + + + , x x a 1 : ( ) ( ) ( ) 1 1 1 1 1 1 1 1 ... ... , n n 1 1 ... n n a a a = + + + . , k x a x a k =1,2,3,... . , 21. ( ) 1 [ ] 1 1 0 n n ... n n f x a x a x a x a x = + + + + , > 0 , 1 + + + < , x x > .

29. 1 1 0 1 1 0 n ... n ... , n n a x a x a a x a x a 1 1 0 ... n n n n a x a x a a x < k = 0,1,2,...,n 1 (2) n a n a > > = . x x k n a a > = , > = , : 1 1 0 n n ... 0 n n a x a x a x a 26 1 1 + + + < + + + x , : 1 + + + < , (1) x 1 , k n k n a x a x n x x > , . n (2), (1) . (2) n k k k n k , 0,1, 2,..., 1 n n , n a k , 0,1, 2,..., 1. n k n k n a . ( ) 1 [ ] 1 1 0 n n ... n n f x a x a x a x a x = + + + + , n a k n k , 0,1, 2,..., 1 n k n a () f (x) > 0 , x x > , f ( x) , () f ( x) , x > , n n a x . () ( ) 1 1 1 1 0 1 1 0 n n ... n n ... 0 n n n n f x a x a x a x a a x a x a x a = + + + + + + + > . () 1 + + + > , x > , f ( x ) = a x n + ( a x n 1 + ... + a x + a ) n n 1 1 0 . ,

30. f (x)[x] , : 22. 1 [ ] 1 1 0 ( ) n n ... , 0 n n n f x a x a x a x a x a = + + + + { } 0 1 1 : max , ,..., . n a a a a = f ( x) : 1 + a ,1 + a a n a n n n n n f ( ) = a + a ... + a + a a ... + a + a a a a n n n n n n n n 2 1 2 1 a a a a a a a a ... ... + + + + + + + + n n 0 1 2 1 0 1 2 1 n n a a a 1 ... 1 , a a > + . (1) + 27 . f ( x) , 1 1 1 1 0 1 1 0 1 ... 1 1 0 0 . n n a a + + + = + = + = >1 ( 2 1 ) 1 1 n + + + + < >1. n 1 n n < , ( ) 1 1 n a a < , : 1 a a n , (1), f (x)., f (x) f (x) 1 a a n . , . , 1 2 , ,..., n , .

31. 0,2,1,0,3,1,2,4 3 , 2 -1, -3 1 2 -4. : 23. ( Budan Fourier) 1 [ ] 1 1 0 ( ) n n ... , 0. f x anx an x a x a x an = + + + + ( ) f ( ), f ( ),..., f (n) ( ) (*) f ( x) , , [ , ], < , , f ( x) , ( ) ( ). ( ) ( ) 2r, r 0. . ( Hariot Descartes) 0 1 2 ... n 0, , 28 2 n i a + a x + a x + + a x = a 0 1 , ,..., n a a a . . f (x) = x6 + 4x5 3x4 x3 + 2x 2 3 , , -2, 2, -1, -3, 4, 1. 3 1. f (x) = x6 4x5 3x4 + x3 2x 2 3 2,2,+1,3,4,+1, f (x) 3 1 . f (x) 3 1 .

32. 1. , , 3 + 3 + 3 = 3 f (x) = x3 + x2 + x + f (1) = 7 . 2. ( x)[x] (x +1) = (x), x. 3. , , f (x) = x3 8x2 8 x + , , = = 1 2 3 1 2 3 . 4. f (x) : (x 2)(x 3), f ( x) x 2 x 3 12 17, . 5. f ( x) (x) = (x )(x ) : ( ) ( ) ( ) ( ) ( ) (i) = + f f f f + = + = 29 , , f f f f x x (ii) ( ) ( ) ( ) ( ) ( ) , , x x 2 2 (iii) ( x) = f ( ) x + ( f ( ) f ( )), =. 6. , f (x) = x +1 + x + ( ) ( )2 x = x 1 . 7. f ( x) x2 + x +1 x2 x +1 x 1 2x +1, . f ( x) x4 + x2 +1. 8. f (x) = ax 1 + x 2 +...+ x +1, a 0, : 1 1 . a < + 9. P(x)[x], : P( x) P(2x2 1) = P(x2 )P(2x 1), x. (1) ( 2002, the monthly problem set)

33. 10. P(x) Q(x) , , P(x2 )+Q( x) = P(x) + x5Q( x) , x. 30 ( 2012) 11. 10 9 8 7 6 5 4 3 2 7 6 5 4 3 2 1 0 P(x) = x 20x +135x + a x + a x + a x + a x + a x + a x + a x + a , 0 1 7 a ,a ,...,a . . ( 2011) 12. a,b, c, d f (x) = ax3 + bx2 + cx + d , : * 1 2 3 + + , ` . 13. ( ) 1 1 1 0 n n ... , f x = a x + a x + + a x + a n n { } 0 , ,..., 1, 1 , n a a a + . 14. ( x) x : (x2 6x +8)( x) = (x2 + 2x)(x 2) . ( 2014)

34. 1. , , 3 + 3 + 3 = 3 f (x) = x3 + x2 + x + f (1) = 7 . 3 + 3 + 3 = 3 : + + = 0 = = . (1) x + 1 = x ax + 1 2 + bx + 1 + c = a x 2 + b x + c ax + a + b x + a + b + c = ax + bx c a = a a + b = b a + b + c = c a = b = c c 31 f (1) = 7 + + = 6 . (2) = = , = + = 6 = = = 2 . + + = 0 , + + = 6 0 = 6, . 2. ( x)[x] (x +1) = (x), x. ( x) = ax2 + bx + c, a,b,c, a 0 , . ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 , 2 , 0, 2 , . . 2, : f (x) = 2cx + c, c . 3. , , f (x) = x3 8x2 8 x + 1 2 3 , , 1 2 3 = = . Vieta : 1 2 3 1 2 2 3 3 1 1 2 3 + + = 8, + + = 8, = , 1 2 3 = = : 2 3 1 1 1 = 8, = 8, = = 512, = 8.

35. 4. f (x) : (x 2)(x 3), f ( x) x 2 x 3 12 17, . : ( ) ( ) ( ) ( ) ( ) ( ) 1 2 f x = x 2 p x +12 f x = x 3 p x +17. : f (2) =12 f (3) =17, , f (x) = ( x 2)(x 3) p(x) + x + = + f f f f + = + = + = + = = = 32 : 2 + = f (2) =12 3 + = f (3) =17 = 5, = 2. f (x) : (x 2)(x 3) (x) = 5x + 2 . 5. f ( x) (x) = (x )(x ) : ( ) f ( ) f ( ) f ( ) f ( ) (i) x x , , (ii) ( ) ( ) ( ) ( ) ( ) , , x x 2 2 (iii) ( x) = f ( ) x + ( f ( ) f ( )), =. (i) f (x) = (x )(x ) (x) + ( x + ), (1) (1) x = x = : ( ) ( ) f ( ) f ( ) f ( ) f ( ) f f , . (ii) = . (i) . (iii) = . ( ) ( )2 ( ) f x = x x + x + , (2)

36. ( ) ( ) ( ) ( )2 ( ) f x = 2 x x + x x + . (3) (2) (3) x = : + = f ( ) f ( ) = = f ( ) = f ( ) f ( ). 6. , f (x) = x +1 + x + ( ) ( )2 x = x 1 . (1 ) : ( ) ( )2 ( ) 1 ( ) ( ) 1 x = x 1 f x = x + + x + f 1 = 0 f 1 = 0, ( ) 1 f x ( ) ( )2 f x : x 1 . f (1) = 01+ + = 0 + = 1, (1) 1 1 1 + + + f x x x x x x x = + + = + = + = + + + + 33 f ( x) ( ) ( ) ( )( 1 ) 1 1 1 x x x x 1 ... 1 , ( ) 1 1 f x = x + x +...x +1+. ( ) 1 f 1 = 0 : +1+ = 0 = ( +1), (1) = . (2 ) 5(iii) 17, : f (1) = 0 f (1) = 01+ + = 0 ( +1) + = 0 = ( +1), = . 7. f ( x) x2 + x +1 x2 x +1 x 1 2x +1, . f ( x) x4 + x2 +1. x4 + x2 +1 = (x2 + x +1)(x2 x +1). , :

37. f (x) = (x4 + x2 +1) p(x) + (x) = (x2 + x +1)(x2 x +1) p(x) + (x), (1) ( x) , (x) = ax3 + bx2 + cx + d, a,b, c, d . x x x x x x x x x 1 1 1 2 1 = + + + + = + + + + x x x x x x + + + + + + = + + + + + + = + = + + + = + = + = + = + = + = + = = + = + = + = = = = x = x + x + x + + x = x + x + x + 34 (1) : f (x) (x) = (x2 + x +1)(x2 x +1) p( x) , f (x) (x) x2 + x +1 x2 x +1. , f ( x) ( x) x2 + x +1 x2 x +1 , , x 1 2x +1, . : ( x) = (x2 + x +1)( x + ) + x 1, (2) ( x) = (x2 x +1)( x + ) + 2x +1. (3) (2) (3) : ( ) ( 2 )( ) ( 2 )( ) ( ) ( ) ( ) ( ) 3 2 3 2 1 1 2 1 , , 1 2, 1 1 , 1, 2 , 2 1, 2 1 , 3 , 1. 2 2 : ( ) ( 2 1) 3 1 3 1 2 3 1 . 2 2 2 2 8. f (x) = ax 1 + x 2 +...+ x +1, a 0, : 1 1 . a < + . 22. 9. P(x)[x], : P( x) P(2x2 1) = P(x2 )P(2x 1), x. (1) ( 2002, the monthly problem set)

38. r P( x)[x], [x] x . : 0 = P(r ) P(2r2 1) = P(r2 )P(2r 1), r2 2r 1 P(x) . : () r >1, r2 > r 2r 1 > r ( r2 2r 1) P(x) , . P(x) 1. () r < 1, r2 >1, () r2 P(x) . 2r 1< r . 2r-1, P(x) , . P(x) -1. P(x) [1,1]. r . = , : + + + + = = : P x P r P r P r r r P x r r r r r + + + > > < 35 x r + r 1, 1 2 0 ( ) ( ) 1 2 1 1 1 2 1 1 ( ). 2 2 2 2 r r r + 1 1, 2 > < ( ), 2 2 1 1 1 1 2 1, 2 2 2 ( ). , P(x) r , . , P(x) , r =1. : ( ) ( 1) ( ), (1) 0. k P x = x Q x Q :

39. ( ) ( )( ) ( ) ( ) ( )( ) ( ) x 1 k Q x 2 x 2 2 k Q 2 x 2 1 x 2 1 k Q x 2 2 x 2 k Q 2 x 1, x = ( ) ( ) ( ) ( ) Q x Q x Q x Q x x 2 1 2 1 = , Q 2x 1 = S (x2 ) = S (x), = S (a) 36 2 2 2 1 = 2 1, , (2) Q(x) (1). P(x) , , Q(x) , 1, , Q(1) 0 . Q(x) , Q(x) 0, x . (2) ( ) ( ) ( ) ( ) 2 2 , Q x Q x x Q x Q x ( ) ( ) ( ) S x Q x x, x x2 : ( ) ( 2 ) ( 4 ) ... ( 2 ), k S x = S x = S x = = S x k . (3) a >1, , (3) ( ) Q ( 2x 1 ) ( ) S x Q x , . S ( x) ( ) ( 2 1 1 ) 1 1 ( 1 ) Q S Q = = , : S (x) =1, x. : Q(x) =Q(2x1), x, Q(x) . Q(x) = c (2) Q(x) . ( ) ( 1) , , . n n n P x = c x n` c 10. P(x) Q(x) , , P(x2 )+Q( x) = P(x) + x5Q( x) , x. ( 2012)

40. P(x2 ) P(x) = (x5 1)Q(x), x. (1) : 1, , 2 , 3 , 4 , P x b x x x x R x = = + + + + + 37 cos 2 sin 2 i = + , 5 5 : 5 =1 6 =, 8 =3. (1) P() = P(2 ),P(2 ) = P(4 ),P(3 ) = P(6 ) = P( ),P(4 ) = P(8 ) = P(3 ), : P( ) = P(2 ) = P(3 ) = P(4 ). b P( ), P(2 ), P(3 ) P(4 ), P( x) b , 2 , 3 , 4 , : ( ) ( )( 2 )( 3 )( 4 ) ( ) ( ) ( ) ( ) 4 3 2 1 . P x x x x x R x b P(x) R(x) b . , P(x) , R(x) . R( x) = 0, , (1) (x5 1)Q(x) = 0 , , [x] , Q(x) = 0 , . R(x) , , R(x) = a 0 . P( x) = a (x4 + x3 + x2 + x +1)+ b = a (x4 + x3 + x2 + x)+ c, a*, c = a + b. , (1)

41. ( 2 ) ( ) ( 5 ) ( ) P x P x = x Q x ( 8 6 4 2 ) ( 4 3 2 ) ( 5 ) ( ) a x x x x a x x x x x Q x + + + + + + = a ( x 8 x 3 x 6 x ) ( x 5 ) Q ( x ) a ( x x )( x ) ( x ) Q ( x ) ( x ) ax ( x ) Qx ( ) + = 3 + 5 = 5 + = 2 5 P x P x = x Q x x 1 P x P x ax ax ax a ax ax ax a x 1 a x x a x a x a x a x x 1 a x x 1 x x a x a x a x a x x 1 a xx 1 ax ax a a x a a x . = + + + + + + + + + + + 4 3 ( ) 2 ( ) P x P x x Q x 1 (1) = x 1 P x P x a x a x a x a x a a x a x a x a x a x 1 a x 1 x x a x 1 x x a x a x a x a x a x a x x 1 x 1 ax ax a a x a a x a a x a a x = + + + + + + + + + + + + + + 38 5 3 1 1 1 1 1 1 0. : Q(x) = a (x3 + x), a* . 2 (1) 5, , min degQ(x) =1 min deg P(x) = 3. ( ) 3 2 3 2 1 0 0 1 2 3 P x = a x + a x + a x + a , a , a ,a , a , 3 a 0, ( ) ( ) ( ) ( ) 1 (1) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 5 2 6 4 2 3 2 3 2 1 0 3 2 1 0 5 6 3 4 2 2 3 2 1 2 1 5 5 3 4 2 2 3 2 1 2 1 5 5 4 3 2 3 2 3 1 2 3 1 2 3 1 2 3 1 a x a x + a a x + a a x = 0 , 2 3 1 2 1 3 1 2 3 a = a = 0, a a = 0, a a = 0a = a = a = 0 , . P(x) , . ( ) 4 3 2 4 3 2 1 0 P x = a x + a x + a x + a x + a , * 0 1 2 3 4 a ,a ,a ,a ,a . , , : ( 2 ) ( ) ( 5 ) ( ) ( 5 ) ( 2 ) ( ) 8 6 4 2 4 3 2 ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) ( ) 4 3 2 1 0 4 3 2 1 0 5 5 3 3 5 4 2 4 3 2 4 3 2 1 4 3 2 1 5 5 3 4 3 2 4 3 2 4 4 3 1 2 3 1

42. ( ) 4 ( ) 3 ( ) 2 ( ) * 2 4 4 3 1 2 3 1 1 2 3 4 a a x + a a x + a a x + a a x = 0a = a = a = a , ( ) 4 3 2 ( 4 3 2 ) * 4 3 2 1 0 4 0 4 0 P x = a x + a x + a x + a x + a = a x + x + x + x + a , a , a . (1) : ( 2 ) ( ) ( 5 )( 3 ) ( 5 ) ( ) ( ) ( 3 ) 4 4 4 P x P x = x 1 a x + a x = x 1 Q x Q x = a x + x , * = = = s x t x u xx , . i i i j i i i j = = + . 0 8 2 10 7 11 i j 11 1 11 9 36 84 126 126 84 36 9 1 20 135 , = = + + + + = + + + + + + + + + 39 4 a . 11. 10 9 8 7 6 5 4 3 2 7 6 5 4 3 2 1 0 P(x) = x 20x +135x + a x + a x + a x + a x + a x + a x + a x + a , 0 1 7 a ,a ,...,a . . 1 2 x , x ,, 9x . 9 9 2 1 1 1 9 = = < Vieta s = 20 ,u = 135 s = 135 20 + 2 t = s2 2u = 130 2 , ( ) ( )( ) 2 i j 1 9 x x t u < 11. = 11 1 2 9 x = x = ... = x = 1 ( ) ( )( ) 9 ( )( 9 8 7 6 5 4 3 2 ) 10 9 8 7 6 5 4 3 2 7 6 5 4 3 2 1 0 P x x x x x x x x x x x x x x x x ax ax ax ax ax ax ax a =11. 12. a,b, c, d f (x) = ax3 + bx2 + cx + d , : * 1 2 3 + + , ` . : b = a ,c = a 2 d = a 3 . :

43. f (x) = ax3 +a x2 + a 2x + a3 = a x2 (x + ) + 2 (x + ) = a(x + )(x2 + 2 ), 1 2 3 = , = i , = i. + + = + + = + + i i i = = + + + = ` ` ` ` = + = + a a = , + + + = 2 (1) a a ... ... 2 2 3, n n i j n n + + + = + + = 40 ( ) ( ) ( ) ( ) ( ( ) ) 1 2 3 * 1 2 3 1 1 1 3 , 4 , , 4 1, . , 4 2, , 4 3, 13. ( ) 1 1 1 0 n n ... , f x = a x + a x + + a x + a n n { } 0 , ,..., 1, 1 , n a a a + . . 1 2 , ,..., n f (x) . , Vieta, : 1 a a 1 2 ... n n n 1 n i j i j n n < , f (x) , : 2 2 2 ( )2 2 1 2 1 2 1 2 1 i < j n 2 2 2 1 2 1 n = . (2) : 2 2 2 1 2 ... n + + + n , n 3. : n =1, f (x) = x +1 f (x) = x 1. n = 2, f ( x) = x2 + x 1 f ( x) = x2 x 1. n = 3, (1) , { } 1 2 3 , , 1,+1 , : f ( x) = ( x 1)(x2 1).

44. 14. ( x) x : (x2 6x +8)( x) = (x2 + 2x)(x 2) . 41 ( 2014) (x 2)(x 4)(x) = x(x + 2)(x 2), x , (1) , x = 0,2 4 : (0) = (2) = (2) = 0 . ( x) x, x + 2 x 2 , : ( x) = x (x + 2)(x 2)Q(x), (2) Q(x) . (2) (1) : ( )( )2 ( ) ( ) ( )( ) ( ) ( ) x 2 x 4 x x + 2 Q x = x x + 2 x 2 x x 4 Q x 2 , (3) x . , x (x + 2)( x 2)( x 4) ( x 2)Q( x) xQ( x 2) = 0, x . (4) (4), x (x 2)( x + 2)(x 4) , : (x 2)Q(x) xQ(x 2) = 0, x . (5) x = 0 Q(0) = 0, Q(x) = xR(x), R(x) . (5) (4) : (x 2) xR(x) x (x 2) R(x 2) = 0 x( x 2) R( x) R(x 2) = 0, , x(x 2) 0( x) , : R(x) = R(x 2) , x . R(x) = R(x + 2k ), k ], x , R(x) x, c = R(0) = R(2k ), k ] . R(x) = c, x , : (x) = x (x + 2)( x 2)Q(x) = x ( x + 2)( x 2) xR( x) = cx2 (x2 4) . 15

45. ( ( )) ( ( )) ( ) ( ) P x 3 + 3 P x 2 = P x3 3P x , x. (1) n n n n x 3 x Q x 3 x Q x Q x 3 x 6 x Q x 3 Q x x Q x Q x x = + + + + + = n n n n n 2 2 3 2 c x c c x c c c x c c c c c c c n mm 3 1 3 2 1 3 2 0, + + + + + + + = 42 ( 2014) deg P(x) = 0 , P(x) = a 0 . (1) : a3 + 3a2 = a 3aa3 + 3a2 + 2a = 0a = 1 a = 2. P( x) = 1 P(x) = 2 . deg P(x) = n > 0. P(x) P( x) = axn +Q( x), a 0 degQ( x) = k n 1 Q(x) = 0 . (1) : ( axn +Q ( x )) 3 + 3 ( axn +Q ( x )) 2 = ax 3 n +Q ( x 3 ) 3 Q ( x ) 3 a ( 1 ) n xn , (2) x, x3n a 0 , : 3 1 1. a a a a = = = : . a =1. (2) : ( ( )) ( ( )) ( ) ( ) ( ) ( ) ( ) 3 3 2 3 3 3 3 1 , (3) xn +Q x + xn +Q x = x n +Q x Q x n xn x = x ( ) ( ) ( ( )) ( ( )) ( ) ( ( )) ( ) ( ) ( ) ( ) 2 2 3 2 2 3 3 3 1 . Q(x) = 0 , : 3 2 3 ( 1) , x n = n xn . degQ( x) = k > 0 , , 0 < k < n , : 2n + k = deg(x) = deg(x) = max{3k, n} 3k n k, . degQ( x) = 0, Q(x) = c 0 . (x) = ( x) : ( ) ( ( ) ) ( ) n ` 2 3 2 * 1 0, 2 1 0, 3 2 0 1, 2 , . + = + + = + + = = = : P(x) = x2m 1, x,m`* . . a = 1. (2) :

46. ( ( )) 3 3 ( ( )) 2 3 ( 3 ) 3 ( ) 3 ( 1 ) xn Q x xn Q x x n Q x Q x n xn + + + = + + = ( x ) x n Qx ( ) x n ( Qx ( )) ( Qx ( )) x n xQx n ( ) ( Qx ( )) ( x ) Q ( x ) Q ( x ) ( ) x 3 3 3 6 3 , = + + + = + n n n n n 2 2 3 2 c x c c x c c c x c c c c c c c n mm 3 1 3 2 1 3 2 0, + + + + + + = 43 ( ) ( ) , x x 2 2 3 2 2 3 3 3 1 . , Q(x) = 0 , : 3 2 3 ( 1) , x n = n xn . degQ( x) = k > 0 , , 0 < k < n , : 2n + k = deg(x) = deg(x) = max{3k, n} 3k n k, . degQ( x) = 0, Q(x) = c 0 . (x) = ( x) : ( ) ( ( ) ) ( ) n 2 3 2 * 1 0, 2 1 0, 3 2 0 1, 2 , . + = + + = + + = = = ` : P( x) = x2m 1, x,m`. : P( x) = 1, P(x) = 2 , P( x) = x2m 1, P(x) = x2m 1, x,m`*.

47. 1. f (x) = xn + 4 ][x], , , n 4. 2. f (x) = xn + 5xn1 + 3 , n >1 . f ( x) . 3. P(x)[x], : P( x) P(2x2 1) = P(x2 ) P(2x 1), x. (1) 4. P(x)[x], : P(x2 ) = P( x) P( x + 2), x. 5. P(x)[x], : P(x2 )+ P(x) = P(x2 ) + P( x), x. 6. P(x)[x], 44 : () 200. () 2. () 4. () P(1) = 0, P(2) = 6 P(3) = 8.

48. ... 2014 1. , f : , ( ) - x . f () = { f ( x) : x}:= Im f f () x y = f (x) 45 f . f f ( x) y = f (x) x. f : , 1 2 x , x 1 2 x < x , ( ) ( ) 1 2 f x f x . f : , 1 2 x , x 1 2 x < x , ( ) ( ) 1 2 f x < f x . f : , 1 2 x , x 1 2 x < x , ( ) ( ) 1 2 f x f x . f : , 1 2 x , x 1 2 x < x , ( ) ( ) 1 2 f x > f x .

49. 2014 f : , - , , , . f : , K,M : K < f ( x) < M, x. f : , M > 0 , : f (x) < M, x. , , , . f : 1 1 (), - 1 2 x , x , - : ( ) ( ) 1 2 1 2 1 2 x , x , x x f x f x , (1) 46 , : ( ) ( ) 1 2 1 2 1 2 x , x , f x = f x x = x . (2) 1 1, (2). f : , : f () = . , - y f (x) = y (3) x . y , (3) , f () f . f , f () = . y f () , (3) , y x , f (x) = y . f 1 - 1 - f , f 1 : f () , y f 1 ( y) := x . f 1 ( y) := x (3) . f : 1 1 (3) , y f () 1 2 x , x f y , ( ) ( ) 1 2 f x = y = f x . f f . - .

50. - , - . 2. 1. f : - 47 x2 f (x) + f (1 x) = 2x x4 , (1) x . . x 1 x (1) ( )2 ( ) ( ) ( ) ( )4 1 x f 1 x + f x = 2 1 x 1 x , (2) (1) (2) f (x) =1 x2 . : , - . . - . 2. f : * * + + , : x2 ( f (x) f ( y)) ( x y) f ( f ( x) y), x, y * . + + = + . f (1) = a , x = y =1 : f (a) = a . x = a, y =1, : 2 3 ( 1) 1 ,0,1 2 a a a a = + , a > 0 a =1 f (1) =1. , x =1, y = t > 0 : f (t ) = 1, t > 0. t 3. f :__ - f (x + y) = f (x) + f ( y) ( Cauchy), (1) x, y_. . (1) x = y = 0 f (0) = f (0) + f (0) , f (0) = 0. f (kx) = kf (x), k `*, x_.

51. 2014 k =1, f (1 x) = f (x) =1 f (x) . k `* , f ((k +1)x) = f (kx + x) = f (kx) + f ( x) = kf ( x) + f ( x) = (k +1) f ( x) . (1) y = x , f (x + (x)) = f (x) + f (x) f (0) = f (x) + f (x) f (x) = f ( x), f (kx) = f (kx) = kf (x) , k `* . : f (kx) = kf (x), k ], x_. (2) = = = = 48 (2) x 1 k f k 1 kf 1 f (1) kf 1 f 1 1 f (1) k k k k k , m],n`* f m f m 1 m f 1 m f (1). n n n n = = = , x_ f (x) = cx, c = f (1) . : f (x) = cx, c = f (1)_ f (x + y) = c ( x + y) = cx + cy = f (x) + f ( y). 2 f : , f _, , - : x ( ) n n * p ` x , ( ) n n * q ` - x. : , * n n p x q n` . 4. f : - f (x + y) = f (x) + f ( y) ( Cauchy), (1) x, y f (x) 0, x 0 . 2, : f (x) = cx, c = f (1), x_ . x y , x y 0 , f (x y) 0 f (x) = f ((x y) + y) = f (x y) + f ( y) f ( y) . f .

52. x , - ( n )n * p ` x ( ) n n * q ` x n n p x q . p x q f p f x f q n n n n f p f x f q pf f x qf = = n + n n + n n + n n + n xf f x xf f x f x cx x = , : = = . 49 (1) ( ) ( ) ( ) (1) n n n n p f = f p f x f q = q f . n + , f (x) = f (1) x = cx, c 0 , x. . 1. 3, f (x) 0, x 0 , - f . f (x) = f (1) x = cx, x. , f , , ( ) n n * p ` x ( ) n n * q ` x . f - : ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) lim lim lim 1 lim 1 1 1 1 , . , 2. f (x) 0, x 0 , f . ( x ) x x : f ( x ) f ( x ) . n n ` * nn ( p ) p x lim p x n n ` * n n n ( ) ( ) lim (1) ( ) ( ) (1) n n n f p f x pf f x f x f x + x. 3. f , - f 0 x . , ( ) n n * x ` nx x : ( ) ( ) n f x f x . , - n n 0 0 y = x x + x x , , f 0 x :

53. 2014 ( ) ( ) ( ) ( ) ( ) ( ) f y f x x x f x f x f x f x = + = + n n n 0 0 0 ( ) ( ) ( ) ( ) f x f x f x f x 0 . n n 4. f : f (x + y) = f ( x) + f ( y) , 0, : f (x) M, x(a, a), a > 0. f 0, , . , ( ) n n * x ` 0. n x 0, n Nx N . = a > 0 , K > 0 , : , n n Nx < a a < Nx < a n > K . f Nx M , n > K Nf x M , n > K n n f x M n K > + + f x M x + + f x f M x M f + + f x M f x 50 : ( ) ( ) ( ) , , n N , > 0, N M N < , : ( ) , n f x < n > K. : lim f ( x ) 0 f (0) n + n = = , f 0. 5. , f - 0 x , f , , 0. . , f ( , ) , , : f (x) M, x( , ), < , : ( ) , , 2 2 2 , , 2 2 2 ( ) , , 2 2 2 2 , f 0. 5. = , [0,+) (0,+) . f : -

54. f (x + y) = f (x) + f ( y) , (1) f (xy) = f (x) f ( y), (2) = = = 51 x, y , : f (x) = 0, x f (x) = x, x . 2, : f (x) = f (1) x , x_ . : f (1) = 0 , f (x) = f (1 x) = f (1) f ( x) = 0 , x . f (1) 0 , f (1) = f (11) = f (1) f (1) f (1) =1, f (x) = x, x_ . , y 0 ( ) ( ) ( ) ( ) ( ) 2 f y = f y y = f y f y = f y 0 f (x + y) = f (x) + f ( y) f (x) , f . , x _ , ( ) n n * p ` x ( ) n n * q ` x n n p x q . f , ( ) ( ) ( ) n n n n p = f p f x q = f q , n + , f (x) = x, , x . 6. f : (0,+) f (xy) = f (x) f ( y), x, y > 0 . f , : f (x) = 0, x > 0 c f (x) = xc , x > 0 . x > 0 ( ) ( ) ( ) 2 f x = f x x = f x 0 . : ( ) ( ) ( ) ( ) 2 f 1 = f 11 = f 1 f 1 = 0 1, : f (1) = 0 , f (x) = f ( x 1) = f (x) f (1) = 0, x > 0 . f (1) =1, f (x) > 0 , x > 0 . , x > 0 f (x) = 0 , f (1) f x 1 f (x) f 1 0, x x .

55. 2014 g : g (w) = ln f (ew ) , - : ( + ) = ( xy + ) = ( xy ) = ( x ) ( y ) g x y f e f e e f e f e ln ln ln ln ln . ( x ) ( y ) ( ) ( ) f e f e g x g y = + = + f , g , : g (w) = cw, w. cw = ln f (ew ) f (ew ) = ecw f (x) = xc , x > 0 . 7 ( 2002). f : 52 ( f (x) + f (z))( f ( y) + f (t )) = f (xy zt ) + f (xt + yz) , x, y, z,t . f (x) = c , x. 4 2 2 0 1 c = cc = c = 2 f (x) = 0 ( ) 1 f x = . 2 f . x = z = 0 2 f (0)( f ( y) + f (t )) = 2 f (0) , f (0) = 0 f ( y) + f (t ) =1. y = t ( ) 1 f y = , , 2 f (0) = 0. y =1, z = 0,t = 0, f (x) f (1) = f (x) f (1) =1, - f ( x) . z = 0,t = 0, f (x) f ( y) = f (xy), x, y , w > 0 ( ) ( )2 f w = f w 0 . x = 0, y =1,t =1 2 f (1) f (z) = f (z) + f (z) , f (z) = f (z), z , f . g : (0,) g (w) = f ( w) 0 . x, y > 0 g (xy) = f ( xy ) = f ( x y ) = f ( x ) f ( y ) = g ( x) g ( y) . (1) f ,

56. g (x2 ) = f (x), x, z = y,t = x ( ( ) ( )) (( ) ) ( ( )) g x2 g y2 2 g x2 y2 2 g x2 y2 2 + = + = + , x, y . a = x2 , b = y2 , xm + , ( ) 0 0 = = = = 53 g (a) + g (b) = g (a + b), a,b > 0 . (2) (1) (2), 4. g (w) = w, w > 0 , f (0) = 0 f f (x) = g (x2 ) = x2 , x. , (x2 + z2 )( y2 + t2 ) = (xy zt )2 + (xt + yz)2 , Lagrange. 8. f : (0,+)(0,+) f (xf ( y)) = yf (x), x, y > 0 f (x)0, x+. x =1 = y f ( f (1)) = f (1) . x =1, y = f (1) ( ( ( ))) ( )2 f f f 1 = f 1 , ( )2 ( ( ( ))) ( ( )) ( ) ( ) f 1 = f f f 1 = f f 1 = f 1 f 1 =1. 1 . y = x f (xf (x)) = xf (x) , xf (x) - , x > 0 . f 0 x >1. ( )0 0 f x = x ( ) 2 0 0 0 x f x = x 2 ( 2 ) 4 0 0 0 0 x f x = x ,...., xm , m = 2k , k `* . 0 x >1 0 f xm = xm + , . f 0 x >1. f x(0,1). 1 f 1 x f 1 f (x) xf 1 f 1 1 x x x x x ,

57. 2014 f 1, . f (0,1) , f 1, xf (x) =1, x > 0 , f (x) 1 , x 0 = > = = = = = x + . 54 = > . x : f (x) 1 , x 0 x f (xf ( y)) f x y y 1 yf (x) y x x f (x) 1 0, x 3. 1. f :]] , : f ( f (x) + y) = x + f ( y + 2009), x, y]. . y = 0 y =1 f ( f (x)) = x + 2009 f ( f (x) +1) = x + f (2010) . (1) : f ( f (x) +1) f ( f (x)) = f (2010) f (2009) f (z +1) f (z) = f (2010) f (2009), zIm f . (2) Im f = ] , (1). (2) f ] , : f (z) = az + b, - a,b] . - : f ( f (x) + y) = a ( f (x) + y) + b = a2x + ay + ab + b , x + f ( y + 2009) = x + a ( y + 2009) + b = x + ay + 2009a + b , x, y]. , , , a2 =1, 2009a = ab(a,b) = (1, 2009) (a,b) = (1, 2009). : f (x) = x + 2009, x]. 2. f :`` , : f (xf ( y)) = yf (x), x, y`. f (2007) .

58. . f 1-1. , f (x1 ) = f (x2 ) , 55 y ` , : ( ) ( ( )) ( ( )) ( ) 1 1 2 2 1 2 x f y = f yf x = f yf x = x f y x = x . x =1 : f ( f ( y)) = yf (1), y`, y =1 : f ( f (1)) = f (1) f 1-1, f (1) =1, (1) y ` : f ( f ( y)) = y (2) (2) f ` . , z` , y = f (z)` , f ( y) = f ( f ( z)) = z . , , x`, z = f ( y) : f (xz) = f (xf ( y)) = yf (x) = f ( z) f ( x) , , * 1 2 , ,..., n x x x ` , n`* , : ( ) ( ) ( ) ( ) 1 2 1 2 , ,...,n n f x x x = f x f x f x . (3) f ( p) - p . , f ( p) = ab, a,b 1, (2) (3) : p = f ( f ( p)) = f (ab) = f (a) f (b) . (4) f 1-1 f (1) =1, (4) f (a) >1 f (b) >1, , p . 2007 = 32 223 , (4) : ( ) ( ( )) ( ) 2 f 2007 = f 3 f 223 , f (3) f (223) . f (3) = 2, f (2) = 3, f (223) 5. f (2007) 22 5 = 20. f (3) = 3, f (223) 2, f (2007) 32 2 =18. f (3) f (223) f (2007) 18. f f (2007) =18. f : x , x = 2k 223m q, k,m q`* (q, 2) =1 (q, 223) =1, f (x) = f (2k 223m q) = 2m 223k q ,

59. 2014 f (2007) = f (20 2231 32 ) = 232 =18 . . , 1 1 1 x = 2k 223m q 2 2 2 y = 2k 223m q : ( ( )) ( k m ( k m )) f xf y f q f q 2 223 2 223 2 223 2 223 , = 1 1 2 2 1 2 = = ( ) k m m k m k m k f 1 + 2 1 + 2 q q 1 + 2 2 + 1 q q + + + = + + + = + = + > : = + + . 56 1 2 1 2 ( ) 2 2 ( 1 1 ) 2 1 2 1 2 1 1 2 yf x = 2k 223m q f 2k 223m q = 2k +m 223m +k q q . f (2007) 18. 3. f : * * + + , : f (xf ( y)) f (xy) x, x, y * . + = + . x f (x) : f ( f (x) f ( y)) = f ( f ( x) y) + f (x). (1) , x, y : f ( yf (x)) = f (xy) + y (2) (1) (2) : f ( f ( x) f ( y)) = f (xy) + y + f ( x) , (3) x, y : f ( f ( x) f ( y)) = f (xy) + x + f ( y) , (4) (3) (4) : y + f (x) = x + f ( y) f (x) x = f ( y) y, (5) x, y > 0. g (x) = f (x) x (0,+), f ( x) = x + c, x > 0, c > 0 . f ( x) c =1, 4. f : *+ , : () f (1) = 2008 () f (x) x2 +10042 () * 1 1 1 1, f x y f x f y x, y . x y y x + . u x 1 , v y 1 , x, y 0 y x f (u + v) = f (u) + f (v) . (1) : uv xy 2 1 4 xy

60. u,v > 0 uv 4 u x 1 , v y 1 , = + = + (2) y x x, y > 0. , (2) x uy v y uy uy uvy v = = + = + = = i x x , + + = + + + = + + + = + + = + + 2 2 2 2 2 2 f x x x f x x 1004 1004 1004 + + 2 2 2 2 x x fx x x x x f x x x x x hx x x 2008 1004 2008 2008 1004 2004 2008 2004 , 0 1004 1004 , 0 + + 2 2 > + 2 2 > h x x 57 2 1 1 2 0 1 y x uy , (3) = ( uv )2 4uv 0 , - , S = v > 0 P = v > 0 . - u y (3), x uy 1 i y ( , ) i i x y (2), u,v > 0, uv 4 . (1) u,v > 0. , u,v > 0, - w > 0 (u + v)w 4, u (v + w) 4, vw 4 . ( ) ( ) ( ) f u v w f u v f w ( ) ( ) ( ) ( ) ( ) ( ) f ( u v ) f ( u ) f ( v ) f u v w f u f v w f u f v f w , u,v 0 . h(x) := f (x) f (1) x = f (x) 2008x , x > 0 , h(x) = 0, x > 0 . : h(x+ y) = f (x+ y) 2008(x+ y) = f (x) + f ( y) 2008x2008y = h(x) +h( y) , x, y > 0 h(x +1) = h(x) + h(1) = h( x), x > 0 . (3) , () : ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ] 2 2 1005 1004 , 0,1 . h(x) (0,1] - (0,+), (3). , h(nx) = nh(x), n h(x) (0,+), h(x) = 0 f (x) = 2008x.

61. 2014 5. f :]] , , p , : () m n(mod p), f (m) = f (n) . () f (mn) = f (m) f (n), m,n] . . n ] : f (0) = f (0) f (n) f (0) = 0 f (0) = f (n) =1. f , : f (n) = 0, n] f (n) =1, n] . (1) f . : 0 = f (0) = f (kp), k ]. = = 58 m f (m) = f (mp ) = f (m)p , f (m){1,0,1}. a (primitive) p , f (a) 0 , f . f (a) = 1. f (a) =1, : p n ( ) 0, / , 1, / . f n p n = (2) f (a) = 1, : ( ) pn n 0, / , 1, / (mod ), 1, / (mod ). f n p n n p p n n p = (3) , f (1), (2) (3). (3) - Legendre. 6. f : , : f (x2 y2 ) = (x y) f (x) + f ( y) , x, y. . x = y = 0 f (0) = 0, y = 0, x f (x2 ) = xf (x), x. (1) x 0 : f ( ( ) x2 ) ( ) f x fx x , f .

62. a x , 0 = = ( ) ( ) , 0, g x f x xg x y xg x yg x y yg y 2 = 2 , 2 = 2 . f x f x f x g x g x = = = = . 59 x x a . - g (x) : f (x) = xg (x) , x. x2 y2g (x2 y2 ) = (x y) xg ( x) + yg ( y) , x, y. x y , (x + y) g (x2 y2 ) = xg (x) + yg ( y), x, y, x y. (2) y y (x y) g (x2 y2 ) = xg (x) yg ( y), x, y, x y. (3) (2) (3), x y , : ( 2 2 ) ( ) ( 2 2 ) ( ) , x 0, y 0, x y : g (x2 y2 ) = g (x), g (x2 y2 ) = g ( y), : g (x) = g ( y), x, y 0, x y. g , x 0, , g (0) = a = g (1) , g (x) = a , x f (x) = ax, x . , f (x) = ax, x . . (1) x 0 : 2 ( ) ( ) ( ) ( ) 2 ( ) 2 2 x x x x = 0 , g (x2 ) = g (x), x . , g - , x ( ) 1, , 1, . h x x = h(2x) = h( x) , g (x) = h(log x), x > 0, g (x2 ) = g ( x), x . 7. f : , : f (x y) = f (xy) + xy + f ( y), x, yIm f .

63. 2014 . f c , c = f (x c) = f (x) + cx + f (c) = cx + 2c, f x = f y y = f y + y y + f y 1 2 1 1 2 2 2 2 1 2 y f y f + + = + + b b f f f + = + = 60 x, c = 0 f = 0. f . a,b f (a) = b 0 , x, : f (x b) = f (x) + bx + f (b) f (x b) f ( x) = bx + f (b). , { f (x b) f (x) : x} = {bx + f (b) : x} = . (1) , bIm f = { f (x) : x} , 2 ( ) ( ) 2 ( ) ( ) b f ( 0 ) 2 f b b f b b f b f b + = + + = . (2) (1) , x , 1 2 y , y Im f 1 2 x = y y , , (2), ( ) ( ) ( ) ( ) ( ) ( ) 1 2 ( ) 2 2 1 2 ( ) ( ) 0 0 2 2 0 0. 2 2 y y y y f x f = + = + , f (b) = f (0 b) = f (0) + 0b + f (b) = f (0) + f (b), b, ( ) 2 2 ( ) ( ) ( ) 0 2 0 0 0, 2 2 f ( x) = 0, x. 8. f : , : (( ( )) ( )) ( ) 2 f f x + f y = xf x + y, x, y. . x , y = xf ( x) ( ) ( ) 2 a = f x + f y , f (a) = 0. x = y = a f (02 + 0) = a 0 + a f (0) = a. x = a, y f ( f ( y)) = y, y. f (x) 1-1 , f (a) = f (b)a = f ( f (a)) = f ( f (b)) = b , y z = f ( y) ,

64. f (z) = f ( f ( y)) = y . x f (x) y = a (( ( ( ))) ( )) ( ) ( ( )) 2 f f f x + f a = f x f f x + a, x, a f (x2 ) = xf ( x) + a, x, a. (1) , x y = a (( ( )) ( )) ( ) 2 f f x + f a = xf x + a, x, a ( ( )2 ) ( ) f f x = xf x + a, x, a. (2) 2 2 2 2 2 2 = = + u v f u v f f u f v f w af ab = + 61 f ( f (x)2 ) = f (x2 ) f ( f ( f (x)2 )) = f ( f (x2 )) f ( x)2 = x2 , x f (x) = x f (x) = x. u,v, uv 0, - f (u) = u f (v) = v , ( ) ( ) ( ( ) ( )) ( ) ( ) 2 2 2 , uf u v u v = + = + u4 2u2v + v2 = u4 + 2u2v + v2 2u2v = 2u2vuv = 0, . , f (x) = x, x f (x) = x, x. 9. f : (0,+)(0,+), - : f ( f (x) + y) = xf (1+ xy), x, y(0,+). (1) 1. f . , a,b 0 < a < b , f (a) < f (b) . bf ( b ) af ( a ) w = > 0 b a , - f (a) < f (b) , w > f (b) > f (a) . , x = a, y = w f (a) : ( ( ) f ( b ) f ( a )) 1 , b a , x = b, y = w f (b)

65. 2014 ( ) ( ( ) ( )) 1 . = + + = + = > , f (x) 1 1 1 + = , . < , f (x) 1 1 1 + = , . = x >1. 1 1 f f x xf x x 62 f b f a f w bf ab b a , a = b, . 0 < a < b f (b) f (a) . 2. x = y =1, (1): f ( f (1) +1) = f (2) . , x =1, y = 2 : f ( f (1) + 2) = f (3), x = 2, y =1, - f ( f (2) +1) = 2 f (3), : 2 f (3)( f (2) +1) = f (( f (1) +1) +1) = ( f (1) +1) f ( f (1) + 2) = ( f (1) +1) f (3) , f (1) +1 = 2, f (1) =1. 3. x >1 y = 1 1 . : x f f (x) 1 1 xf (1 x 1) xf ( x) x . f (x) 1 x + > , - x f f f (x) 1 1 f (1) 1 x . xf (x) 1 f (x) 1 x f (x) 1 x + < , - x f f f (x) 1 1 f (1) 1 x . xf (x) 1 f (x) 1 x f (x) 1 , x x > 0 . f (x) +1 >1, : ( ( ) ) ( ) 1 f f x f x + = + .. (2) y =1 (1) ( 1+ x >1) ( ( ) 1) (1 ) 1 x + = + = + . (3) (2) (3) : 1 , ( ) = 1 1 x f x + + x x > 0 , f (x) 1 , = x > 0 . x

66. 10. f : (1,+) , t = , (4) 1 (2) f = f , (1) : t > t : ( ) ( ) 2 f t f f t t > t . 5 x = y = , f (x) c , 63 : f (x) f ( y) = ( y x) f ( xy), x, y(1,+). (x, y) = (t, 2),(t, 4) (2t, 2) - : f (t) f (2) =(2t) f (2t), f (t) f (4) =(4t) f (4t), f (2t) f (2) =(22t) f (4t) , : f (4) + (t 3) f (2) = t (2t 5) f (4t ), t >1. (1) (1) 5 2 2 5 (2) (2 5) (4 ), 2 t f = t t f t t >1, , 1 5 2 4 f t = 2 f t t ( ) ( ) ( ) ( ) (2) (4 ) (2) 2 ( ) 4 4 4 , 2 2 f t f t f t t t = + = + = 1 5 2 t = , 2 5, 2 2 = x c = 2 f (2). 11. f : , : f (x) f ( y) = f (xy +1) + f (x y) 2, x, y. (1) y = 0 : f (x) f (0) 1 = f (1) 2 . (0) ( ) ( 1 ) 2 f 1, ( ) () 0 1 f f x c f = = , - : c2 = c + c 2c2 2c + 2 = 0, . f (0) =1. , f (1) 2, 0 f (x) = f (1) 2 0 , . f (1) = 2 . (1) x y : f (x) f ( y) = f (xy +1) + f ( y x) 2, x, y , (1) f (x y) = f ( y x), x, y . , y = 0, f .

67. 2014 , (1), y y , f (x) f (y) = f (xy +1) + f (x + y) 2, x, y f (x) f ( y) = f (1 xy) + f ( x + y) 2, x, y. (2) (1) (2) : f (x + y) f (x y) = f (1+ xy) f (1 xy), x, y. (3) x m n y m n m n + (3) , , , = = ( ) ( ) f m f n f m n f m n = + g x = f + x f x x g r s + g s t = f r f s + f s f t g x 2 + g y 2 + = g x 2 y 2 + xy + + + g x 2 xy + y 2 + 1 1 2 1 1 2 1 2 g x g y g x y g a g b g ab a b 64 2 2 2 2 2 2 1 1 4 4 . ( ) 1 1 , 4 4 , : g (m2 n2 ) = f (m) f (n), m,n. (4) n = 0 , (4) : f (m) = g (m2 )+ f (0) = g (m2 )+1, m (5) f (1) = 2 , g (1) =1. , (4) : ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 , , , . f r f t g r t r s t = = , m = r2 s2 , n = s2 t2 : g (m) + g (n) = g (m+ n) , g (0) = 0 g . f (m) = g (m2 )+1, m, - , : ( ) ( ) ( ) ( ) ( 2 ) ( 2 ) ( 2 2 ) ( ) ( ) ( ) , , 0. = = > g , : g (a) g (b) = g (a) g (b) = g (ab) = g ((a)b) , g (a) g (b) = g (a) g (b) = g (ab) = g (a (b)) g (a) g (b) = g (a) g (b) = g (ab) , g . g , g (x) = x, x 5). f (x) = g (x2 )+1 = x2 +1, x. 12. P(x)[x], :

68. P( x) P(2x2 1) = P(x2 )P(2x 1), x. (1) ( 2002, the monthly problem set) r P(x)[x], [x] - x .. - : 0 = P(r ) P(2r2 1) = P(r2 )P(2r 1), r2 2r 1 P(x) . : () r >1, r2 > r 2r 1 > r - ( r2 2r 1) P(x) , . - P(x) 1. () r < 1, r2 >1, () r2 P(x) . 2r 1< r . 2r-1, - P(x) , . P(x) -1. P(x) [1,1]. r . = , : + + + + = = P x P r P r P r r r P x r r r r r + + + > > < x 1 k Q x 2 x 2 2 k Q 2 x 2 1 x 2 1 k Q x 2 2 x 2 k Q 2 x 1, x = Q x Q x Q x Q x x 65 x r + r 1, 1 2 0 ( ) ( ) 1 2 1 1 1 2 1 1 ( ). 2 2 2 2 : r r r + 1 1, 2 > < , 2 2 1 1 1 1 2 1, 2 2 2 . , P(x) r , . - , P(x) , r =1. : ( ) ( 1) ( ), (1) 0. k P x = x Q x Q : ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( 2 ) ( 2 ) ( ) 2 1 = 2 1, , (2) Q(x) (1).

69. 2014 P(x) , , - Q(x) , 1, , Q(1) 0 . Q(x) , Q(x) 0, x . (2) 2 1 2 1 = , Q 2x 1 = : = S (a) 66 ( ) ( ) ( ) ( ) 2 2 , Q x Q x x Q x Q x ( ) ( ) ( ) S x Q x S (x2 ) = S (x), x, x x2 : ( ) ( 2 ) ( 4 ) ... ( 2 ), k S x = S x = S x = = S x k . (3) a >1, , (3) ( ) Q ( 2x 1 ) ( ) S x Q x , . S (x) ( ) ( 2 1 1 ) 1 1 ( 1 ) Q S Q = = , : S (x) =1, x. : Q(x) =Q(2x1), x, Q(x) . Q(x) = c (2) - Q(x) . ( ) ( 1) , , . n n n P x = c x n` c 4. 13. f : , : f (x2 + f ( y)) = ( x y)2 f ( x + y), x, y. 14. f : , - : f (x + y) = f (x) + f ( y) + 2xy, x, y. 15. f : , : f ( f (x) + y) = 2x + f ( f ( y) x), x, y. 16. f : , : f (x2 ) f ( y2 ) = ( x + y) f (x) f ( y) , x, y.

70. 17. f : , : () f (x + y) = f (x) + f ( y), x, y. () f (x) x2 f 1 , 67 = x x* . 18. f : , : f ( y + zf (x)) = f ( y) + xf (z), x, y, z. 19. f : , : f (xf (z) + f ( y)) = zf (x) + y, x, y, z. 20. f : * * + + _ _ , : () f (x 1) f (x) 1, x * . + + = + _ () ( ) ( ) f x3 = f x 3 , x *+ _ . 21. f : , : f (x + y) f ( f (x) y) = xf ( x) + yf ( y), x, y. 22. f :`` f (m+ f (n)) = f ( f (m)) + f (n) , m,n`. ( 1996) 23. f :`` f ( f (n)) = n +1987 , n`. ( 1987) 24. f : f (x) f ( y) = f ( f ( y)) + xf ( y) + f ( x) 1, x, y . ( 1999) 25. P(x)[x], : P(x2 ) = P(x) P(x + 2), x. 26. P(x)[x], : P(x2 )+ P(x) = P(x2 )+ P(x), x. 27. P(x)[x], - : () 200. () 2. () 4. () P(1) = 0, P(2) = 6 P(3) = 8.

71. 2014 28. f : f (x + yf (x)) = f (x) + xf ( y) , 68 x, y . 29. f , g : f (x + yg (x)) = g (x) + xf ( y) , x, y . 30. f : f ([x] y) = f (x) f ( y) , x, y . ( 2010) 31. f , , a b ( ) a, f(b) f(b + f(a) 1). ( , ) ( 2009) 32. f : f (xf (x + y)) = f ( yf (x)) + x2 , x, y . 33. f : + + _ _ f ( f (x)2 y) = x3 f (xy), x, y + _ , _+ .

72. ... 2014 ... - . - . , 2 = {(x, y) : x, y}, , 2 ( ) ( ) ( ) 1 1 2 2 1 2 1 2 x , y + x , y := x + x , y + y [], ( ) ( ) ( ) 1 1 2 2 1 2 1 2 1 2 2 1 x , y x , y := x x y y , x y + x y []. :, x (x) := (x,0). (0,1) ( )2 ( ) ( ) ( ) 0,1 := 0,1 0,1 = 1,0 1, i , i2 = 1, z2 +1 = 0 . - (x,0) = x (x, y) = (x,0) + (0, y) = (x,0) + y (0,1) x + yi . (0,0) 0 + 0i 0 . 69

73. 2014 - ( ) 2 2 z ( ) 1 2 z + z = z z , z 0. z z 70 f :, z = x + yi f (z) :=(x, y) . - . - . (z) 1 2 z = x + yi z = x yi . z , M x . - : 1 2 1 2 z + z = z + z 1 2 1 2 z z = z z , 1 2 1 2 z z = z z 1 1 2 2 2 z = x + yi . z = OM = x2 + y2 . (1) : 2 z z = z ( ) 1 1 z (0,0) ( ) 2 1 z z O(0,0)

74. = , 2 z 0 z 0 , x z z + z y z z z = = = = . (2) = = . (3) 71 z1 z2 = z1 z2 1 1 z z z z = 1 1 2 2 z z 1 2 1 2 z + z z + z ( ), 1 2 1 2 z z z + z z = x + yi , x z , y - z . x = Re z y = Imz . z - : Re Im i 2 2 z = x + yi 0 + 0i z = r (cos + i sin ) , r = x2 + y2 = z z , cos x , sin y r r (3), + 2k , k . (3) z arg z , , [0, 2 ), z rg z . 0 . , z =1+ i z = 2 ,

75. 2014 cos 1 , sin 1 2 , 2 2 4 z = + i 1 = = = + z i i = + = + > = y i + y > = = + > z = r z r cos + + i sin + , = + , 72 k k = = = + , z i = + z =1+ i 2 cos sin 4 4 . , 1 3 2 2 cos 1 sin 3 2 2 k , k , 2 2 3 1 cos 2 sin 2 cos 2 sin 2 3 3 3 3 . z = x > 0 = 0 , z = x (cos 0 + i sin 0) , z = x < 0 = z = x (cos + i sin ). z = yi ( ) y , 0, arg 2 3 , 0, 2 yi y < o cos sin , 0 2 2 cos 3 sin 3 , 0 2 2 z yi y i y . ( ) ( ) ( ) 1 1 1 1 2 2 2 2 z = r cos + i sin , z = r cos + i sin , z = r cos + i sin , , : ( ) ( ) 1 2 12 1 2 1 2 z z = r r cos + + sin + , 1 1 ( ) ( ) 1 2 1 2 2 2 1 1 cos ( ) i sin ( ) z r

76. zn = rn (cos n + i sin n ), n ( De Moivre). , modulo 2 , , - , . 1 2 z , z 0, arg ( z z ) = arg ( z ) + arg ( z ) ( mod 2 ) , 1 2 1 2 arg z 1 arg ( z ) arg ( z ) ( mod 2 ) = = > 73 1 2 2 z = , arg 1 arg z (mod 2 ) z = , z 0 , arg (zn ) = narg z (mod 2 ), z 0, n. ( ) ( ) 1 1 2 2 z , z ( ) 1 2 1 2 1 2 d , = = z z . 1: (a),(b) - , (,) - , ( ). (z) , : 1. (z)((a),(b)) . 2. : z a (b z) z a 0 b z . 3. t(0,1) z = (1 t )a + tb . 4. arg (z a) = arg (b z)(mod 2 ) . 1 2 . (z)((a),(b)) a z + z b = a b z a = (b z), > 0 23

77. 2014 z a b z z a = = + b z = t a + tb + + = = 74 t = , > 0 1 + , t(0,1) 0 1 t t = > , - : ( ) 1 (1 ) 1 1 . 34 arg (z a) arg (b z)(mod 2 ) arg z a 0(mod 2 ) b z z a = (b z), > 0 . [,] , ( - , - . ( : (z)( z a 0 arg (z a) arg (b a)(mod 2 ) b a = > = . . : : (a),(b) : 1. (z) 2. z a b a 3. t z = (1 t )a + tb . 4. 0 z a z a b a b a = . 5. 1 1 0 1 z z a a b b = . . 1,2 3 - 1. :

78. z a z a z a z a z a b a b a b a b a b a = = 0 z z z a z a z a b z z a b z a b + = . M OM OM ,OM - M OM . 1 2 75 z a z a b a b a = 1 0 z a z a 1 0 1 0 0 1 0 a a a a b a b a b b b a b a = = = . . (a) (b) - . (z) 1, : = . ( ) 1 + = = + > 0 , , < 0 , - . =1 - 2 ( ) 1 1 M z ( ) 2 2 M z - . ( ) 1 2 1 2 , OM 1 ( ) . . M OM , 1 2 2 1 M OM 3

79. 2014 M OM xOM xOM arg z arg z arg z (mod 2 ) = = = . M OM - M OM M OM arg z mod 2 arg z mod 2 = = = z z M OM M OM z 2 2 arg (mod 2 ) z = = z z z z arg (mod 2 ) arg (mod 2 ). = = z i i i i z i M OM i arg = = . arg 1 arg 3 M M M arg z z M M M ( 4) 2 3 M M M = M O M = arg z z 76 2 1 2 2 1 2 1 z 1 2 1 : 2 ( ) 1 ( ) 2 1 1 2 1 2 2 1 M OM 2 2 1 1 2 1 2 1 1 2 , ( ) ( ) 1 1 2 2 M z M z , 1 2 z =1 i, z =1+ i , ( )2 2 1 1 + 1 + 2 1 2 2 = = = = , 1 2 2 ( ) 2 1 2 M OM i i = = = ( ) ( ) ( ) 1 1 2 2 3 3 M z ,M z M z M M M . 2 1 3 3 1 2 1 3 z z 2 1 = . , 1 M O, 2 1 3 MOM , O 2 3 M,M 2 1 3 1 z z , z z , . 3 1 2 1 3 2 3 z z 2 1 .

80. , M1 (3+ 2i),M2 (4 + 3i),M3 (2 + 2i) - 1 2 3 z = 3+ 2i, z = 4 + 3i, z = 2 + 2i M M M z z i arg arg 1 arg 1 = = = = z z i arg arg 1 = = + = arg z z 0 z z 77 3 1 2 1 3 2 1 1 2 4 + 2 1 ( ) 3 1 2 3 1 4 M M M z z i z z . ( ) ( ) ( ) 1 1 2 2 3 3 M z ,M z M z . ( 5): 1 2 3 M ,M ,M 3 1 2 1 = z z z z 3 1 * 2 1 = . : z z = z , : z z z z z z z z M ,M ,M 3 1 = 2 1 1 2 3 3 1 2 1 . 3 4 1 M 2 M 3 M 2 M 3 M 1 M 2 M O O

81. 2014 1 M 2 M O O 6 M M M M M M M z z 2 M 3 arg 3 z z 2 2 2 2 , . z z i z z z z z z z z z z = = z z z z z z z z z z z z 78 ( 6): 3 1 2 1 3 1 2 1 3 2 1 3 1 * 3 1 2 1 2 1 3 1 2 1 = = = = (cross-ratio) - , ( ) ( ) ( ) ( ) 1 1 2 2 3 3 4 4 M z ,M z ,M z M z - [ ] 1 3 1 4 1 3 2 4 1 2 3 4 2 3 1 4 2 3 2 4 : : : : z z z z z z z z . ( ) ( ) ( ) 1 1 2 2 3 3 M z ,M z ,M z ( ) 4 4 M z . . , , - , . , , z z z z ( ) ( ) ( ) 2 2 3 3 4 4 M z ,M z M z , 2 3 2 4 . 3 M 3 M 1 M 5

82. ( ) 1 1 M z - z z z z , , 1 3 1 4 z : z : z : z : z z z z = z z z z 79 . . M1 (z1 ),M2 (z2 ),M3 (z3 ) M4 (z4 ) - . C , ( ) ( ) ( ) 2 2 3 3 4 4 M z ,M z M z . ( ) 1 1 M z C , , 3 1 4 3 2 4 M M M M M M - . arg z z arg z z 0 4 1 4 2 3 1 3 2 z z z z = [ ] 1 3 2 4 * 1 2 3 4 z z z z 1 4 2 3 . , ( ) ( ) 1 1 2 2 M z M z - ( ) ( ) 3 3 4 4 M z M z , - . , ( ) ( ) ( ) ( ) 1 1 2 2 3 3 4 4 M z ,M z ,M z M z , - - . , , : ( ) ( ) ( ) ( ) 1 1 2 2 3 3 4 4 M z ,M z ,M z M z - , : z z z z z z z z 1 3 2 4 * 1 4 2 3 , 1 3 * 1 4 z z z z 2 3 * 2 4 .

83. 2014 - M1 (z1 ),M2 (z2 ),M3 (z3 ) M4 (z4 ) , : a b a = b = ab a b c a b a c a ab c a abc aab c a b a c a 80 z z z z 1 3 * 1 4 z z z z , 2 3 * 2 4 . z =1, A(a) : a = 1 aa = 1 a = 1 . a 7 8 A(a), B(b) , - : 1 1 a b . C(c) AB , : . c a abc b c a b c ab = = = + + = + = A O O B

84. O One Two ThreeInfinity G. Gamow. , . . - = , =, , . - , . . - , . z , a , a . z + a , = ( ) = 90 , - 81 A B C 9

85. 2014 i (z + a) . = + , , a + i (z + a) . + + 2 x 2 (a) (z) ( a + i (z + a) (ai) 10 A(-a) , ( ) , a i z a a i z a 82 (a-i(z-a)) = 90 , a i (z a). ( ) ( ) 2 ai + + + = , , - z , .

86. ... 2014 ... , . , . . . 1. () , . . 1 83

87. 2014 () , , . , , , : , , . 84 2 &= &= & = () , , : & & + = . 2 2 () , , : & & = . 2 , . () , , , , , , , ,. , , = , = , = , =

88. 3 1. , , , 85 + = . 2 . 4 , , :

89. 2014 = & . (1) + = = + , 86 2 2 , (1), : + = 2 . (2) , = 2 & (3) (2) (3) : + = , , . & , & , . 2. . , . (1 ) 5 : = . , , . , . , , . , : = ( ). = = ( ).

90. = 3600 ( + + ) = 3600 450 ( + ) = = , = = = = . = + + = + + 180 180 90 180 90 90 . = + = + + = + = 87 = 2700 ( + ) = 900 + = + + = 450 + + 450 = 900 + , : = . = . , . : = + = + = = 900 , . 2 , , . : , 2 2 , : 2 2 6 , = 900 + = 900 + = 900 + = . + =+, = . ( 0 ) 0 ( 0 ) 0 0 0

91. 2014 3. = . , , , . (1 ) , , . , : & & & & , (1) = = = = . (2) 2 2 2 = = . (3) 7 (1) 1 1 = . 88 = = , (3) 1 1 2 . , & & , (4) , & & . (5)

92. 8 = . (6) 89 1 . 1 (5) . 1 = (7) = = , = . (6) (7) = . 1 1 3 , , . , &= &= , . , 9

93. 2014 : = , = = , = , . , : &= & &= & , &= , . 90 , = . (1) & & : = = . (2) (1) (2) : = .

94. B- -

95. 1 - 1 . 1 , . (1 : ) . 9 2 // = . 2 = . ( ). = // . = , = // . : = // 2 = // 2 // = . (2 : ) . : = + = + = . 2 2 2 = . 2 2 // = 1 . 2 , . : = = .

96. .. 2014 , , , . = , = ( , ). 9 3 = , + + = , = . , , . , ( ) , , . , , = ( , ).

97. 9 4 = ( ) = . , , , , . , , , , . 1 . 3 , . 30o , . , . ( = 90o ) , = . = = ( ) . 1 . 4 () . () .

98. .. 2014 () . () . , . : 90o 1 = 1 = 1 2 = ( ). 1 2 = . = , . () . , . , : = 90o , , : // (1) . , : = 90o , , : // ( 2 ) . (1) ( 2 ) , . 9 5