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

2009 Croatia Team Selection Test, 4
Tags: modular arithmetic , number theory unsolved , number theory
Determine all triplets off positive integers $ (a,b,c)$ for which $ \mid2^a\minus{}b^c\mid\equal{}1$

2007 Pre-Preparation Course Examination, 8
Tags: number theory unsolved , number theory
Let $m,n,k$ be positive integers and $1+m+n \sqrt 3=(2+ \sqrt 3)^{2k+1}$. Prove that $m$ is a perfect square.

1984 IMO Longlists, 66
Tags: modular arithmetic , number theory unsolved , number theory , IMO Shortlist
Let $1=d_1<d_2<....<d_k=n$ be all different divisors of positive integer n written in ascending order. Determine all n such that: \[d_6^{2} +d_7^{2} - 1=n\]

1995 Greece National Olympiad, 1
Tags: number theory unsolved , number theory
Find all positive integers $n$ such that $-5^4 + 5^5 + 5^n$ is a perfect square. Do the same for $2^4 + 2^7 + 2^n.$

2015 JBMO Shortlist, NT5
Tags: number theory , prime numbers , modular arithmetic , number theory proposed , number theory unsolved
Check if there exists positive integers $ a, b$ and prime number $p$ such that $a^3-b^3=4p^2$

2005 MOP Homework, 1
Tags: number theory unsolved , number theory
Find all triples $(x,y,z)$ such that $x^2+y^2+z^2=2^{2004}$.

2002 Vietnam National Olympiad, 2
Tags: quadratics , inequalities , number theory unsolved , number theory
Determine for which $ n$ positive integer the equation: $ a \plus{} b \plus{} c \plus{} d \equal{} n \sqrt {abcd}$ has positive integer solutions.

2010 Indonesia TST, 4
Tags: number theory unsolved , number theory
How many natural numbers $(a,b,n)$ with $ gcd(a,b)=1$ and $ n>1 $ such that the equation \[ x^{an} +y^{bn} = 2^{2010} \] has natural numbers solution $ (x,y) $

2010 Contests, 4
Tags: algorithm , algebra , system of equations , number theory unsolved , number theory
The sequence of Fibonnaci's numbers if defined from the two first digits $f_1=f_2=1$ and the formula $f_{n+2}=f_{n+1}+f_n$, $\forall n \in N$. [b](a)[/b] Prove that $f_{2010} $ is divisible by $10$. [b](b)[/b] Is $f_{1005}$ divisible by $4$? Albanian National Mathematical Olympiad 2010---12 GRADE Question 4.

2011 Turkey MO (2nd round), 4
Tags: LaTeX , modular arithmetic , number theory unsolved , number theory , legendre s theorem
$a_{1}=5$ and $a_{n+1}=a_{n}^{3}-2a_{n}^{2}+2$ for all $n\geq1$. $p$ is a prime such that $p=3(mod 4)$ and $p|a_{2011}+1$. Show that $p=3$.

2009 Germany Team Selection Test, 2
Tags: induction , number theory , prime numbers , number theory unsolved
Let $ \left(a_n \right)_{n \in \mathbb{N}}$ defined by $ a_1 \equal{} 1,$ and $ a_{n \plus{} 1} \equal{} a^4_n \minus{} a^3_n \plus{} 2a^2_n \plus{} 1$ for $ n \geq 1.$ Show that there is an infinite number of primes $ p$ such that none of the $ a_n$ is divisible by $ p.$

2004 Korea National Olympiad, 2
Tags: algebra , polynomial , number theory , relatively prime , number theory unsolved
$x$ and $y$ are positive and relatively prime and $z$ is an integer. They satisfy $(5z-4x)(5z-4y)=25xy$. Show that at least one of $10z+x+y$ or quotient of this number divided by $3$ is a square number (i.e. prove that $10z+x+y$ or integer part of $\frac{10z+x+y}{3}$ is a square number).

2010 Malaysia National Olympiad, 6
Tags: number theory unsolved , number theory
A two-digit integer is divided by the sum of its digits. Find the largest remainder that can occur.

2013 Dutch IMO TST, 4
Tags: modular arithmetic , number theory unsolved , number theory
Determine all positive integers $n\ge 2$ satisfying $i+j\equiv\binom ni +\binom nj \pmod{2}$ for all $i$ and $j$ with $0\le i\le j\le n$.

2003 China Team Selection Test, 2
Tags: modular arithmetic , quadratics , number theory unsolved , number theory
Positive integer $n$ cannot be divided by $2$ and $3$, there are no nonnegative integers $a$ and $b$ such that $|2^a-3^b|=n$. Find the minimum value of $n$.

1991 Polish MO Finals, 3
Tags: number theory unsolved , number theory
Define \[ N=\sum\limits_{k=1}^{60}e_k k^{k^k} \] where $e_k \in \{-1, 1\}$ for each $k$. Prove that $N$ cannot be the fifth power of an integer.

2005 MOP Homework, 4
Tags: modular arithmetic , algebra , binomial theorem , number theory , number theory unsolved
Let $p$ be an odd prime. Prove that \[\sum^{p-1}_{k=1} k^{2p-1} \equiv \frac{p(p+1)}{2}\pmod{p^2}.\]

2008 Finnish National High School Mathematics Competition, 3
Tags: modular arithmetic , number theory unsolved , number theory
Solve the diophantine equation \[x^{2008}- y^{2008} = 2^{2009}.\]

2007 Estonia National Olympiad, 3
Tags: number theory unsolved , number theory
Prove that the sum of the squares of any three pairwise different positive odd integers can be represented as the sum of the squares of six (not necessarily different) positive integers.

1999 IberoAmerican, 1
Tags: geometry , 3D geometry , number theory unsolved , number theory
Find all the positive integers less than 1000 such that the cube of the sum of its digits is equal to the square of such integer.

2011 Iran Team Selection Test, 2
Tags: number theory , least common multiple , arithmetic sequence , number theory unsolved
Find all natural numbers $n$ greater than $2$ such that there exist $n$ natural numbers $a_{1},a_{2},\ldots,a_{n}$ such that they are not all equal, and the sequence $a_{1}a_{2},a_{2}a_{3},\ldots,a_{n}a_{1}$ forms an arithmetic progression with nonzero common difference.

1999 Bulgaria National Olympiad, 1
Tags: number theory unsolved , number theory
Find the number of all integers $n$ with $4\le n\le 1023$ which contain no three consecutive equal digits in their binary representations.

2012 Turkmenistan National Math Olympiad, 6
Tags: number theory unsolved , number theory
Prove that $1^{2011}+2^{2011}+3^{2011}+...+2012^{2011} $ is divisible by $2025078$.

2008 Regional Competition For Advanced Students, 4
Tags: floor function , logarithms , modular arithmetic , number theory unsolved , number theory
For every positive integer $ n$ let \[ a_n\equal{}\sum_{k\equal{}n}^{2n}\frac{(2k\plus{}1)^n}{k}\] Show that there exists no $ n$, for which $ a_n$ is a non-negative integer.

2014 Indonesia MO Shortlist, N6
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.