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Lusophon Mathematical Olympiad
2022 Lusophon Mathematical Olympiad
3
3
Part of
2022 Lusophon Mathematical Olympiad
Problems
(1)
Prove that x is a perfect square
Source: Lusophon Mathematical Olympiad 2022 Problem 3
11/3/2022
The positive integers
x
x
x
and
y
y
y
are such that
x
2022
+
x
+
y
2
x^{2022}+x+y^2
x
2022
+
x
+
y
2
is divisible by
x
y
xy
x
y
.a) Give an example of such integers
x
x
x
and
y
y
y
, with
x
>
y
x>y
x
>
y
.b) Prove that
x
x
x
is a perfect square.
number theory
Divisibility