This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 1362

2013 India Regional Mathematical Olympiad, 2
Tags: number theory unsolved , number theory
Find all triples $(p,q,r)$ of primes such that $pq=r+1$ and $2(p^2+q^2)=r^2+1$.

2011 Polish MO Finals, 1
Tags: number theory unsolved , number theory
Find all integers $n\geq 1$ such that there exists a permutation $(a_1,a_2,...,a_n)$ of $(1,2,...,n)$ such that $a_1+a_2+...+a_k$ is divisible by $k$ for $k=1,2,...,n$

2013 Kazakhstan National Olympiad, 1
Tags: invariant , number theory unsolved , number theory , easy , modular arithmetic
On the board written numbers from 1 to 25 . Bob can pick any three of them say $a,b,c$ and replace by $a^3+b^3+c^3$ . Prove that last number on the board can not be $2013^3$.

2014 Spain Mathematical Olympiad, 2
Tags: modular arithmetic , number theory unsolved , number theory
Let $M$ be the set of all integers in the form of $a^2+13b^2$, where $a$ and $b$ are distinct itnegers. i) Prove that the product of any two elements of $M$ is also an element of $M$. ii) Determine, reasonably, if there exist infinite pairs of integers $(x,y)$ so that $x+y\not\in M$ but $x^{13}+y^{13}\in M$.

2008 Indonesia TST, 4
Tags: number theory , greatest common divisor , number theory unsolved
Let $ a $ and $ b $ be natural numbers with property $ gcd(a,b)=1 $ . Find the least natural number $ k $ such that for every natural number $ r \ge k $ , there exist natural numbers $ m,n >1 $ in such a way that the number $ m^a n^b $ has exactly $ r+1 $ positive divisors.

2004 Brazil National Olympiad, 5
Tags: induction , modular arithmetic , algebra , polynomial , number theory unsolved , number theory
Consider the sequence $(a_n)_{n\in \mathbb{N}}$ with $a_0=a_1=a_2=a_3=1$ and $a_na_{n-4}=a_{n-1}a_{n-3} + a^2_{n-2}$. Prove that all the terms of this sequence are integer numbers.

2003 Iran MO (3rd Round), 1
Tags: Gauss , number theory , relatively prime , number theory unsolved
suppose this equation: x <sup>2</sup> +y <sup>2</sup> +z <sup>2</sup> =w <sup>2</sup> . show that the solution of this equation ( if w,z have same parity) are in this form: x=2d(XZ-YW), y=2d(XW+YZ),z=d(X <sup>2</sup> +Y <sup>2</sup> -Z <sup>2</sup> -W <sup>2</sup> ),w=d(X <sup>2</sup> +Y <sup>2</sup> +Z <sup>2</sup> +W <sup>2</sup> )

2014 Saudi Arabia IMO TST, 3
Tags: induction , algebra , system of equations , number theory unsolved , number theory
Show that it is possible to write a $n \times n$ array of non-negative numbers (not necessarily distinct) such that the sums of entries on each row and each column are pairwise distinct perfect squares.

1979 IMO Longlists, 77
Tags: number theory unsolved , number theory
By $h(n)$, where $n$ is an integer greater than $1$, let us denote the greatest prime divisor of the number $n$. Are there infinitely many numbers $n$ for which $h(n) < h(n+1)< h(n+2)$ holds?

2008 India National Olympiad, 3
Tags: number theory unsolved , number theory , india
Let $ A$ be a set of real numbers such that $ A$ has at least four elements. Suppose $ A$ has the property that $ a^2 \plus{} bc$ is a rational number for all distinct numbers $ a,b,c$ in $ A$. Prove that there exists a positive integer $ M$ such that $ a\sqrt{M}$ is a rational number for every $ a$ in $ A$.

1985 IMO Longlists, 25
Tags: number theory unsolved , number theory
Find eight positive integers $n_1, n_2, \dots , n_8$ with the following property: For every integer $k$, $-1985 \leq k \leq 1985$, there are eight integers $a_1, a_2, \dots, a_8$, each belonging to the set $\{-1, 0, 1\}$, such that $k=\sum_{i=1}^{8} a_i n_i .$

2014 Postal Coaching, 5
Tags: number theory unsolved , number theory
Let $p>3$ be a prime and let $1+\frac 12 +\frac 13 +\cdots+\frac 1p=\frac ab$.Prove that $p^4$ divides $ap-b$.

2011 China Team Selection Test, 2
Tags: induction , number theory unsolved , number theory
Let $\{b_n\}_{n\geq 1}^{\infty}$ be a sequence of positive integers. The sequence $\{a_n\}_{n\geq 1}^{\infty}$ is defined as follows: $a_1$ is a fixed positive integer and \[a_{n+1}=a_n^{b_n}+1 ,\qquad \forall n\geq 1.\] Find all positive integers $m\geq 3$ with the following property: If the sequence $\{a_n\mod m\}_{n\geq 1 }^{\infty}$ is eventually periodic, then there exist positive integers $q,u,v$ with $2\leq q\leq m-1$, such that the sequence $\{b_{v+ut}\mod q\}_{t\geq 1}^{\infty}$ is purely periodic.

1987 Federal Competition For Advanced Students, P2, 4
Tags: number theory unsolved , number theory
Find all triples $ (x,y,z)$ of natural numbers satisfying $ 2xz\equal{}y^2$ and $ x\plus{}z\equal{}1987$.

2014 Indonesia MO, 4
Tags: modular arithmetic , number theory unsolved , number theory
A positive integer is called [i]beautiful[/i] if it can be represented in the form $\dfrac{x^2+y^2}{x+y}$ for two distinct positive integers $x,y$. A positive integer that is not beautiful is [i]ugly[/i]. a) Prove that $2014$ is a product of a beautiful number and an ugly number. b) Prove that the product of two ugly numbers is also ugly.

1996 Irish Math Olympiad, 3
Tags: number theory unsolved , number theory
Suppose that $ p$ is a prime number and $ a$ and $ n$ positive integers such that: $ 2^p\plus{}3^p\equal{}a^n$. Prove that $ n\equal{}1$.

2014 Postal Coaching, 1
Tags: number theory unsolved , number theory
Suppose $p,q,r$ are three distinct primes such that $rp^3+p^2+p=2rq^2+q^2+q$. Find all possible values of $pqr$.

2003 Bulgaria National Olympiad, 2
Tags: inequalities , LaTeX , number theory unsolved , number theory
Let $a,b,c$ be rational numbers such that $a+b+c$ and $a^2+b^2+c^2$ are [b]equal[/b] integers. Prove that the number $abc$ can be written as the ratio of a perfect cube and a perfect square which are relatively prime.

1998 Baltic Way, 3
Tags: number theory unsolved , number theory
Find all positive integer solutions to $2x^2+5y^2=11(xy-11)$.

2000 Irish Math Olympiad, 1
Tags: number theory unsolved , number theory
Consider the set $ S$ of all numbers of the form $ a(n)\equal{}n^2\plus{}n\plus{}1, n \in \mathbb{N}.$ Show that the product $ a(n)a(n\plus{}1)$ is in $ S$ for all $ n \in \mathbb{N}$ and give an example of two elements $ s,t$ of $ S$ such that $ s,t \notin S$.

2010 Contests, 1
Tags: inequalities , number theory , number theory unsolved
Suppose $a$, $b$, $c$, and $d$ are distinct positive integers such that $a^b$ divides $b^c$, $b^c$ divides $c^d$, and $c^d$ divides $d^a$. [list](a) Is it possible to determine which of the numbers $a$, $b$, $c$, $d$ is the smallest? (b) Is it possible to determine which of the numbers $a$, $b$, $c$, $d$ is the largest?[/list]

2005 German National Olympiad, 4
Tags: number theory unsolved , number theory
I am not a spammer, at least, this is the way I use to think about myself, and thus I will not open a new thread for the following problem from today's DeMO exam: Let Q(n) denote the sum of the digits of a positive integer n. Prove that $Q\left(Q\left(Q\left(2005^{2005}\right)\right)\right)=7$. [[b]EDIT:[/b] Since this post was split into a new thread, I comment: The problem is completely analogous to the problem posted at http://www.mathlinks.ro/Forum/viewtopic.php?t=31409 , with the only difference that you have to consider the number $2005^{2005}$ instead of $4444^{4444}$.] Darij

2009 China Girls Math Olympiad, 1
Tags: inequalities , number theory , Diophantine equation , number theory unsolved
Show that there are only finitely many triples $ (x,y,z)$ of positive integers satisfying the equation $ abc\equal{}2009(a\plus{}b\plus{}c).$

1990 China Team Selection Test, 3
Tags: modular arithmetic , induction , number theory unsolved , number theory
Prove that for every integer power of 2, there exists a multiple of it with all digits (in decimal expression) not zero.

2011 Kazakhstan National Olympiad, 2
Tags: number theory unsolved , number theory , Kazakhstan
Determine the smallest possible number $n> 1$ such that there exist positive integers $a_{1}, a_{2}, \ldots, a_{n}$ for which ${a_{1}}^{2}+\cdots +{a_{n}}^{2}\mid (a_{1}+\cdots +a_{n})^{2}-1$.