Olympiad problems

2013 Olympic Revenge, 4

Find all triples $(p,n,k)$ of positive integers, where $p$ is a Fermat's Prime, satisfying \[p^n + n = (n+1)^k\]. [i]Observation: a Fermat's Prime is a prime number of the form $2^{\alpha} + 1$, for $\alpha$ positive integer.[/i]

2015 Irish Math Olympiad, 3

Tags: number theory , perfect cube

Find all positive integers $n$ for which both $837 + n$ and $837 - n$ are cubes of positive integers.

1996 Canada National Olympiad, 5

Tags: function , number theory

Let $r_1$, $r_2$, $\ldots$, $r_m$ be a given set of $m$ positive rational numbers such that $\sum_{k=1}^m r_k = 1$. Define the function $f$ by $f(n)= n-\sum_{k=1}^m \: [r_k n]$ for each positive integer $n$. Determine the minimum and maximum values of $f(n)$. Here ${\ [ x ]}$ denotes the greatest integer less than or equal to $x$.

1999 AIME Problems, 10

Tags: probability , number theory , relatively prime

Ten points in the plane are given, with no three collinear. Four distinct segments joining pairs of these points are chosen at random, all such segments being equally likely. The probability that some three of the segments form a triangle whose vertices are among the ten given points is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

2004 Mediterranean Mathematics Olympiad, 1

Tags: factorial , inequalities , prime factorization , number theory

Find all natural numbers $m$ such that \[1! \cdot 3! \cdot 5! \cdots (2m-1)! = \biggl( \frac{m(m+1)}{2}\biggr) !.\]

2010 ELMO Shortlist, 3

Tags: quadratic , diophantine equation , number theory

Prove that there are infinitely many quadruples of integers $(a,b,c,d)$ such that \begin{align*} a^2 + b^2 + 3 &= 4ab\\ c^2 + d^2 + 3 &= 4cd\\ 4c^3 - 3c &= a \end{align*} [i]Travis Hance.[/i]

2014 Taiwan TST Round 2, 2

Tags: function , number theory , functional equation

Determine all functions $f: \mathbb{Q} \rightarrow \mathbb{Z} $ satisfying \[ f \left( \frac{f(x)+a} {b}\right) = f \left( \frac{x+a}{b} \right) \] for all $x \in \mathbb{Q}$, $a \in \mathbb{Z}$, and $b \in \mathbb{Z}_{>0}$. (Here, $\mathbb{Z}_{>0}$ denotes the set of positive integers.)