Olympiad problems

2001 Italy TST, 3

Find all pairs $ (p, q)$ of prime numbers such that $ p$ divides $ 5^q \plus{} 1$ and $ q$ divides $ 5^p \plus{} 1$.

2014 Kazakhstan National Olympiad, 1

Tags: number theory unsolved , number theory , Kazakhstan

$a_1,a_2,...,a_{2014}$ is a permutation of $1,2,3,...,2014$. What is the greatest number of perfect squares can have a set ${ a_1^2+a_2,a_2^2+a_3,a_3^2+a_4,...,a_{2013}^2+a_{2014},a_{2014}^2+a_1 }?$

2005 Korea - Final Round, 6

Tags: induction , graph theory , number theory unsolved , number theory

A set $P$ consists of $2005$ distinct prime numbers. Let $A$ be the set of all possible products of $1002$ elements of $P$ , and $B$ be the set of all products of $1003$ elements of $P$ . Find a one-to-one correspondance $f$ from $A$ to $B$ with the property that $a$ divides $f (a)$ for all $a \in A.$

2000 Hong kong National Olympiad, 3

Tags: number theory unsolved , number theory

Find all prime numbers $p$ and $q$ such that $\frac{(7^{p}-2^{p})(7^{q}-2^{q})}{pq}$ is an integer.

2013 Kazakhstan National Olympiad, 2

Tags: modular arithmetic , number theory , relatively prime , number theory unsolved , Kazakhstan

Let for natural numbers $a,b,c$ and any natural $n$ we have that $(abc)^n$ divides $ ((a^n-1)(b^n-1)(c^n-1)+1)^3$. Prove that then $a=b=c$.

2007 Pre-Preparation Course Examination, 19

Tags: function , induction , number theory , prime numbers , number theory unsolved

Find all functions $f : \mathbb N \to \mathbb N$ such that: i) $f^{2000}(m)=f(m)$ for all $m \in \mathbb N$, ii) $f(mn)=\dfrac{f(m)f(n)}{f(\gcd(m,n))}$, for all $m,n\in \mathbb N$, and iii) $f(m)=1$ if and only if $m=1$.

2007 China Team Selection Test, 2

Tags: floor function , number theory unsolved , number theory

A rational number $ x$ is called [i]good[/i] if it satisfies: $ x\equal{}\frac{p}{q}>1$ with $ p$, $ q$ being positive integers, $ \gcd (p,q)\equal{}1$ and there exists constant numbers $ \alpha$, $ N$ such that for any integer $ n\geq N$, \[ |\{x^n\}\minus{}\alpha|\leq\dfrac{1}{2(p\plus{}q)}\] Find all the good numbers.

2005 Morocco National Olympiad, 2

Tags: quadratics , induction , number theory unsolved , number theory

Find all the positive integers $x,y,z$ satisfiing : $x^{2}+y^{2}+z^{2}=2xyz$

1984 IMO Longlists, 53

Tags: number theory unsolved , number theory

Find a sequence of natural numbers $a_i$ such that $a_i = \displaystyle\sum_{r=1}^{i+4} d_r$, where $d_r \neq d_s$ for $r \neq s$ and $d_r$ divides $a_i$.

2011 BMO TST, 4

Tags: modular arithmetic , number theory , number theory unsolved

Find all prime numbers p such that $2^p+p^2 $ is also a prime number.

2002 South africa National Olympiad, 6

Tags: number theory unsolved , number theory

Find all rational numbers $a$, $b$, $c$ and $d$ such that \[ 8a^2 - 3b^2 + 5c^2 + 16d^2 - 10ab + 42cd + 18a + 22b - 2c - 54d = 42, \] \[ 15a^2 - 3b^2 + 21c^2 - 5d^2 + 4ab +32cd - 28a + 14b - 54c - 52d = -22. \]

2003 Polish MO Finals, 4

Tags: modular arithmetic , number theory , number theory unsolved

A prime number $p$ and integers $x, y, z$ with $0 < x < y < z < p$ are given. Show that if the numbers $x^3, y^3, z^3$ give the same remainder when divided by $p$, then $x^2 + y^2 + z^2$ is divisible by $x + y + z.$

1988 Polish MO Finals, 2

Tags: induction , pigeonhole principle , modular arithmetic , number theory unsolved , number theory

The sequence $a_1, a_2, a_3, ...$ is defined by $a_1 = a_2 = a_3 = 1$, $a_{n+3} = a_{n+2}a_{n+1} + a_n$. Show that for any positive integer $r$ we can find $s$ such that $a_s$ is a multiple of $r$.

1997 China National Olympiad, 3

Tags: number theory unsolved , number theory

Prove that there are infinitely many natural numbers $n$ such that we can divide $1,2,\ldots ,3n$ into three sequences $(a_n),(b_n)$ and $(c_n)$, with $n$ terms in each, satisfying the following conditions: i) $a_1+b_1+c_1= a_2+b_2+c_2=\ldots =a_n+b_n+c_n$ and $a_1+b_1+c_1$ is divisible by $6$; ii) $a_1+a_2+\ldots +a_n= b_1+b_2+\ldots +b_n=c_1+c_2+\ldots +c_n,$ and $a_1+a_2+\ldots +a_n$ is divisible by $6$.

2000 IberoAmerican, 2

Tags: induction , number theory unsolved , number theory

There are a buch of 2000 stones. Two players play alternatively, following the next rules: ($a$)On each turn, the player can take 1, 2, 3, 4 or 5 stones [b]of[/b] the bunch. ($b$) On each turn, the player has forbidden to take the exact same amount of stones that the other player took just before of him in the last play. The loser is the player who can't make a valid play. Determine which player has winning strategy and give such strategy.

2004 Tournament Of Towns, 5

Tags: induction , number theory unsolved , number theory

For which values of N is it possible to write numbers from 1 to N in some order so that for any group of two or more consecutive numbers, the arithmetic mean of these numbers is not whole?

2006 India Regional Mathematical Olympiad, 6

Tags: search , number theory unsolved , number theory

Prove that there are infinitely many positive integers $ n$ such that $ n(n\plus{}1)$ can be represented as a sum of two positive squares in at least two different ways. (Here $ a^{2}\plus{}b^{2}$ and $ b^{2}\plus{}a^{2}$ are considered as the same representation.)

2004 China Girls Math Olympiad, 7

Tags: number theory unsolved , number theory

Let $ p$ and $ q$ be two coprime positive integers, and $ n$ be a non-negative integer. Determine the number of integers that can be written in the form $ ip \plus{} jq$, where $ i$ and $ j$ are non-negative integers with $ i \plus{} j \leq n$.

2007 Estonia Math Open Junior Contests, 3

Tags: number theory unsolved , number theory

Find all positive integers N with at most 4 digits such that the number obtained by reversing the order of digits of N is divisible by N and differs from N.

2006 China Team Selection Test, 3

Tags: number theory unsolved , number theory

For a positive integer $M$, if there exist integers $a$, $b$, $c$ and $d$ so that: \[ M \leq a < b \leq c < d \leq M+49, \qquad ad=bc \] then we call $M$ a GOOD number, if not then $M$ is BAD. Please find the greatest GOOD number and the smallest BAD number.

2009 Federal Competition For Advanced Students, P2, 4

Tags: number theory unsolved , number theory

Let $ a$ be a positive integer. Consider the sequence $ (a_n)$ defined as $ a_0\equal{}a$ and $ a_n\equal{}a_{n\minus{}1}\plus{}40^{n!}$ for $ n > 0$. Prove that the sequence $ (a_n)$ has infinitely many numbers divisible by $ 2009$.

2012 Traian Lălescu, 3

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

There are $n$ natural numbers written on a blackboard, where $n\in\mathbb{N},\ n\geq 2$. During each step two chosen numbers $a,b$, having the property that none of them divides the other, are replaced by their greatest common divisor and least common multiple. Prove that after a number of steps, all the numbers on the blackboard cease modifying. Prove that the respective number of steps is at most $(n-1)!$.

2012 Poland - Second Round, 3

Tags: number theory , least common multiple , vector , absolute value , number theory unsolved

Let $m,n\in\mathbb{Z_{+}}$ be such numbers that set $\{1,2,\ldots,n\}$ contains exactly $m$ different prime numbers. Prove that if we choose any $m+1$ different numbers from $\{1,2,\ldots,n\}$ then we can find number from $m+1$ choosen numbers, which divide product of other $m$ numbers.

2004 Postal Coaching, 14

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

Find the greatest common divisor of all number in the set $( a^{41} - a | a \in \mathbb{N} and \geq 2 )$ . What is your guess if 41 is replaced by a natural number $n$

1984 USAMO, 2

Tags: AMC , USA(J)MO , USAMO , number theory unsolved , number theory

The geometric mean of any set of $m$ non-negative numbers is the $m$-th root of their product. $\quad (\text{i})\quad$ For which positive integers $n$ is there a finite set $S_n$ of $n$ distinct positive integers such that the geometric mean of any subset of $S_n$ is an integer? $\quad (\text{ii})\quad$ Is there an infinite set $S$ of distinct positive integers such that the geometric mean of any finite subset of $S$ is an integer?