HMMT 二月 2021 · ALGNT 赛 · 第 4 题

HMMT February 2021 — ALGNT Round — Problem 4

Suppose that P ( x, y, z ) is a homogeneous degree 4 polynomial in three variables such that P ( a, b, c ) = P ( b, c, a ) and P ( a, a, b ) = 0 for all real a , b , and c . If P (1 , 2 , 3) = 1, compute P (2 , 4 , 8). 4 Note: P ( x, y, z ) is a homogeneous degree 4 polynomial if it satisfies P ( ka, kb, kc ) = k P ( a, b, c ) for all real k, a, b, c .

解析

Suppose that P ( x, y, z ) is a homogeneous degree 4 polynomial in three variables such that P ( a, b, c ) = P ( b, c, a ) and P ( a, a, b ) = 0 for all real a , b , and c . If P (1 , 2 , 3) = 1, compute P (2 , 4 , 8). 4 Note: P ( x, y, z ) is a homogeneous degree 4 polynomial if it satisfies P ( ka, kb, kc ) = k P ( a, b, c ) for all real k, a, b, c . Proposed by: Milan Haiman Answer: 56 Solution: Since P ( a, a, b ) = 0, ( x − y ) is a factor of P , which means ( y − z ) and ( z − x ) are also factors by the symmetry of the polynomial. So, P ( x, y, z ) ( x − y )( y − z )( z − x ) is a symmetric homogeneous degree 1 polynomial, so it must be k ( x + y + z ) for some real k . So, the answer is P (2 , 4 , 8) (2 + 4 + 8)(2 − 4)(4 − 8)(8 − 2) = = 56 . P (1 , 2 , 3) (1 + 2 + 3)(1 − 2)(2 − 3)(3 − 1)