HMMT 二月 2023 · ALGNT 赛 · 第 4 题

HMMT February 2023 — ALGNT Round — Problem 4

Suppose P ( x ) is a polynomial with real coefficients such that P ( t ) = P (1) t + P ( P (1)) t + P ( P ( P (1))) for all real numbers t . Compute the largest possible value of P ( P ( P ( P (1)))).

解析

Suppose P ( x ) is a polynomial with real coefficients such that P ( t ) = P (1) t + P ( P (1)) t + P ( P ( P (1))) for all real numbers t . Compute the largest possible value of P ( P ( P ( P (1)))). Proposed by: Raymond Feng Answer: 1/9 2 Solution: Let ( a, b, c ) := ( P (1) , P ( P (1)) , P ( P ( P (1)))), so P ( t ) = at + bt + c and we wish to maximize P ( c ). Then we have that a = P (1) = a + b + c, 3 b = P ( a ) = a + ab + c, 2 2 c = P ( b ) = ab + b + c. 2 The first equation implies c = − b . The third equation implies b ( a + 1) = 0, so a = − 1 or b = 0. If 1 1 3 b = 0, then ( a, b, c ) = (0 , 0 , 0). If a = − 1, then b = ( − 1) + ( − 1) b + ( − b ) or b = − , so c = and 3 3 1 1 ( a, b, c ) = ( − 1 , − , ). 3 3 1 1 1 1 The first tuple gives P ( c ) = 0, while the second tuple gives P ( c ) = − − + = , which is the 2 2 3 3 3 9 answer.