PUMaC 2022 · 代数（A 组） · 第 2 题

PUMaC 2022 — Algebra (Division A) — Problem 2

Let P ( x, y ) be a polynomial with real coefficients in the variables x, y that is not identically zero. Suppose that P ( ⌊ 2 a ⌋ , ⌊ 3 a ⌋ ) = 0 for all real numbers a . If P has the minimum possible 2 2 degree and the coefficient of the monomial y is 4, find the coefficient of x y in P . m n (The degree of a monomial x y is m + n . The degree of a polynomial P ( x, y ) is then the maximum degree of any of its monomials.)

解析

Let P ( x, y ) be a polynomial with real coefficients in the variables x, y that is not identically zero. Suppose that P ( ⌊ 2 a ⌋ , ⌊ 3 a ⌋ ) = 0 for all real numbers a . If P has the minimum possible 2 2 degree and the coefficient of the monomial y is 4, find the coefficient of x y in P . m n (The degree of a monomial x y is m + n . The degree of a polynomial P ( x, y ) is then the maximum degree of any of its monomials.) Proposed by Sunay Joshi Answer: 216 Note that the possible values for the pair ( ⌊ 2 x ⌋ , ⌊ 3 x ⌋ ) are (2 k, 3 k ) , (2 k, 3 k + 1) , (2 k + 1 , 3 k +