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Problem 1187

AIME late

Number theory Difficulty 5.2 Prove it HMMT February · United States · 2020

Find the number of ordered pairs of positive integers $(x, y)$ with $x, y \leq 2020$ such that $3x^{2} + 10xy + 3y^{2}$ is the power of some prime.

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

Solution:

We can factor as $(3x + y)(x + 3y)$ . If $x \geq y$ , we need $\frac{3x + y}{x + 3y} \in \{1, 2\}$ to be an integer. So we get the case where $x = y$ , in which we need both to be a power of $2$ , or the case $x = 5y$ , in which case we need $y$ to be a power of $2$ . This gives us $11 + 9 + 9 = 29$ solutions, where we account for $y = 5x$ as well.

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