Olympiad problems

2024 Indonesia Regional, 4

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]

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?

modular arithmetic quadratic reciprocity number theory perfect square perfect squares quadratic residues

1996 IMO Shortlist, 2

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?

modular arithmetic quadratic reciprocity number theory perfect square perfect squares quadratic residues

2018 Romania Team Selection Tests, 2

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.

sum of squares quadratic reciprocity number theory composite number modular arithmetic

2017 Hong Kong TST, 4

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 $n$.

quadratic reciprocity number theory modular arithmetic

2008 Romania Team Selection Test, 3

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

quadratic residue modular arithmetic quadratic reciprocity number theory casework factorization prime factorization quadratic residues