Olympiad problems

1996 IMO, 4

The positive integers $ a$ and $ b$ are such that the numbers $ 15a \plus{} 16b$ and $ 16a \minus{} 15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?

1996 IMO Shortlist, 2

Tags: modular arithmetic , perfect square , number theory , quadratic reciprocity

The positive integers $ a$ and $ b$ are such that the numbers $ 15a \plus{} 16b$ and $ 16a \minus{} 15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?

2008 Romania Team Selection Test, 3

Tags: quadratic reciprocity , quadratic residue , modular arithmetic , number theory

Let $ m,\ n \geq 3$ be positive odd integers. Prove that $ 2^{m}\minus{}1$ doesn't divide $ 3^{n}\minus{}1$.

2017 Hong Kong TST, 4

Tags: number theory , quadratic reciprocity

Let $n$ be a positive integer with the following property: $2^n-1$ divides a number of the form $m^2+81$, where $m$ is a positive integer. Find all possible $n$.

2018 Romania Team Selection Tests, 2

Tags: number theory , quadratic reciprocity , sum of squares , composite number

Show that a number $n(n+1)$ where $n$ is positive integer is the sum of 2 numbers $k(k+1)$ and $m(m+1)$ where $m$ and $k$ are positive integers if and only if the number $2n^2+2n+1$ is composite.

2024 Indonesia Regional, 4

Tags: number theory , quadratic reciprocity , quadratic residue

Find the number of positive integer pairs $1\leqslant a,b \leqslant 2027$ that satisfy \[ 2027 \mid a^6+b^5+b^2.\] (Note: For integers $a$ and $b$, the notation $a \mid b$ means that there is an integer $c$ such that $ac=b$.) [i]Proposed by Valentio Iverson, Indonesia[/i]