Olympiad problems

2024 European Mathematical Cup, 1

We call a pair of distinct numbers $(a, b)$ a [i]binary pair[/i] if $ab+1$ is a power of two. Given a set $S$ of $n$ positive integers, what is the maximum possible numbers of binary pairs in S?

2000 France Team Selection Test, 3

Tags: number theory , greatest common divisor , number theory unsolved , Zsigmondy

Find all nonnegative integers $x,y,z$ such that $(x+1)^{y+1} + 1= (x+2)^{z+1}$.

2014 Contests, 3

Tags: modular arithmetic , number theory unsolved , Zsigmondy

Find all nonnegative integer numbers such that $7^x- 2 \cdot 5^y = -1$

2023 Romania EGMO TST, P2

Tags: number theory , prime divisors , prime numbers , Divisibility , IMO Shortlist , power of 2 , Zsigmondy

Let $ b, m, n$ be positive integers such that $ b > 1$ and $ m \neq n.$ Prove that if $ b^m \minus{} 1$ and $ b^n \minus{} 1$ have the same prime divisors, then $ b \plus{} 1$ is a power of 2.

2005 China Team Selection Test, 1

Tags: number theory , prime divisors , prime numbers , Divisibility , IMO Shortlist , power of 2 , Zsigmondy

Let $ b, m, n$ be positive integers such that $ b > 1$ and $ m \neq n.$ Prove that if $ b^m \minus{} 1$ and $ b^n \minus{} 1$ have the same prime divisors, then $ b \plus{} 1$ is a power of 2.

2014 Bosnia Herzegovina Team Selection Test, 3

Tags: modular arithmetic , number theory unsolved , Zsigmondy

Find all nonnegative integer numbers such that $7^x- 2 \cdot 5^y = -1$

1997 IMO Shortlist, 14

Tags: number theory , prime divisors , prime numbers , Divisibility , IMO Shortlist , power of 2 , Zsigmondy

Let $ b, m, n$ be positive integers such that $ b > 1$ and $ m \neq n.$ Prove that if $ b^m \minus{} 1$ and $ b^n \minus{} 1$ have the same prime divisors, then $ b \plus{} 1$ is a power of 2.

2000 IMO Shortlist, 4

Tags: algebra , polynomial , number theory , Divisibility , IMO Shortlist , Zsigmondy

Find all triplets of positive integers $ (a,m,n)$ such that $ a^m \plus{} 1 \mid (a \plus{} 1)^n$.

2000 France Team Selection Test, 3

Tags: number theory , greatest common divisor , number theory unsolved , Zsigmondy

Find all nonnegative integers $x,y,z$ such that $(x+1)^{y+1} + 1= (x+2)^{z+1}$.

2005 China Team Selection Test, 1

Tags: number theory , prime divisors , prime numbers , Divisibility , IMO Shortlist , power of 2 , Zsigmondy

Let $ b, m, n$ be positive integers such that $ b > 1$ and $ m \neq n.$ Prove that if $ b^m \minus{} 1$ and $ b^n \minus{} 1$ have the same prime divisors, then $ b \plus{} 1$ is a power of 2.