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Prove that x is a perfect square

Source: Lusophon Mathematical Olympiad 2022 Problem 3

November 3, 2022

number theoryDivisibility

Problem Statement

The positive integers

x

and

y

are such that

x^{2022}+x+y^2

is divisible by

xy

.
a) Give an example of such integers

x

and

y

, with

x>y

.
b) Prove that

x

is a perfect square.