Problem:
P ( x ) = 24 x 24 + β j = 1 23 ( 24 β j ) ( x 24 β j + x 24 + j ) P ( x ) = 2 4 x 2 4 + j = 1 β 2 3 β ( 2 4 β j ) ( x 2 4 β j + x 2 4 + j )
Let z 1 , z 2 , β¦ , z r z 1 β , z 2 β , β¦ , z r β P ( x ) P ( x ) z k 2 = a k + b k i z k 2 β = a k β + b k β i k = 1 , 2 , β¦ , r k = 1 , 2 , β¦ , r i = β 1 i = β 1 β a k a k β b k b k β
β k = 1 r β£ b k β£ = m + n p k = 1 β r β β£ b k β β£ = m + n p β
where m , n m , n p p p p m + n + p m + n + p
Solution:
Note that
P ( x ) = x + 2 x 2 + 3 x 3 + β― + 24 x 24 + 23 x 25 + 22 x 26 + β― + 2 x 46 + x 47 P ( x ) = x + 2 x 2 + 3 x 3 + β― + 2 4 x 2 4 + 2 3 x 2 5 + 2 2 x 2 6 + β― + 2 x 4 6 + x 4 7
and
x P ( x ) = x 2 + 2 x 3 + 3 x 4 + β― + 24 x 25 + 23 x 26 + β― + 2 x 47 + x 48 x P ( x ) = x 2 + 2 x 3 + 3 x 4 + β― + 2 4 x 2 5 + 2 3 x 2 6 + β― + 2 x 4 7 + x 4 8
So
( 1 β x ) P ( x ) = x + x 2 + β― + x 24 β ( x 25 + x 26 + β― + x 47 + x 48 ) = ( 1 β x 24 ) ( x + x 2 + β― + x 24 ) ( 1 β x ) P ( x ) = x + x 2 + β― + x 2 4 β ( x 2 5 + x 2 6 + β― + x 4 7 + x 4 8 ) = ( 1 β x 2 4 ) ( x + x 2 + β― + x 2 4 ) β
Then, for x β 1 x ξ = 1
P ( x ) = x 24 β 1 x β 1 x ( 1 + x + β― + x 23 ) = x ( x 24 β 1 x β 1 ) 2 (*) P ( x ) = x β 1 x 2 4 β 1 β x ( 1 + x + β― + x 2 3 ) = x ( x β 1 x 2 4 β 1 β ) 2 β ( * )
One zero of P ( x ) P ( x ) 0 0 P ( x ) P ( x ) ( x 24 β 1 ) 2 ( x 2 4 β 1 ) 2 1 1 ( x 24 β 1 ) 2 ( x 2 4 β 1 ) 2 x 24 β 1 x 2 4 β 1 P ( x ) P ( x ) z k = z k β = 15 k β 1 5 k β k = 1 , 2 , 3 , β― β , 23 k = 1 , 2 , 3 , β― , 2 3 30 k β 3 0 k β
β k = 1 23 β£ sin β‘ 30 k β β£ = 4 β k = 1 5 β£ sin β‘ 30 k β β£ = 4 ( 2 β
( 1 / 2 ) + 2 β
( 3 / 2 ) + 1 ) = 8 + 4 3 k = 1 β 2 3 β β£ sin 3 0 k β β£ = 4 k = 1 β 5 β β£ sin 3 0 k β β£ = 4 ( 2 β
( 1 / 2 ) + 2 β
( 3 β / 2 ) + 1 ) = 8 + 4 3 β
Thus m + n + p = 15 m + n + p = 1 5 β
Note: the expression ( β ) ( β )
( 1 + x + x 2 + β― + x n ) 2 = 1 + 2 x + 3 x 2 + β― + ( n + 1 ) x n + β― + 3 x 2 n β 2 + 2 x 2 n β 1 + x 2 n ( 1 + x + x 2 + β― + x n ) 2 = 1 + 2 x + 3 x 2 + β― + ( n + 1 ) x n + β― + 3 x 2 n β 2 + 2 x 2 n β 1 + x 2 n
The problems on this page are the property of the MAA's American Mathematics Competitions