HMMT 二月 2020 · 冲刺赛 · 第 11 题

HMMT February 2020 — Guts Round — Problem 11

[6] Find the number of ordered pairs of positive integers ( x, y ) with x, y ≤ 2020 such that 3 x +10 xy +3 y is the power of some prime.

解析

[6] Find the number of ordered pairs of positive integers ( x, y ) with x, y ≤ 2020 such that 3 x + 10 xy + 2 3 y is the power of some prime. Proposed by: James Lin Answer: 29 3 x + y Solution: We can factor as (3 x + y )( x + 3 y ). If x ≥ y , we need ∈ { 1 , 2 } to be an integer. So we x +3 y get the case where x = y , in which we need both to be a power of 2, or the case x = 5 y , in which case we need y to be a power of 2. This gives us 11 + 9 + 9 = 29 solutions, where we account for y = 5 x as well.