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: 1766

2012 India IMO Training Camp, 2
Tags: modular arithmetic , inequalities , number theory proposed , number theory
Let $0<x<y<z<p$ be integers where $p$ is a prime. Prove that the following statements are equivalent: $(a) x^3\equiv y^3\pmod p\text{ and }x^3\equiv z^3\pmod p$ $(b) y^2\equiv zx\pmod p\text{ and }z^2\equiv xy\pmod p$

2005 Iran Team Selection Test, 1
Tags: function , number theory proposed , number theory
Find all $f : N \longmapsto N$ that there exist $k \in N$ and a prime $p$ that: $\forall n \geq k \ f(n+p)=f(n)$ and also if $m \mid n$ then $f(m+1) \mid f(n)+1$

1982 IMO Longlists, 23
Tags: number theory proposed , number theory
Determine the sum of all positive integers whose digits (in base ten) form either a strictly increasing or a strictly decreasing sequence.

2005 Baltic Way, 20
Tags: number theory proposed , number theory
Find all positive integers $n=p_1p_2 \cdots p_k$ which divide $(p_1+1)(p_2+1)\cdots (p_k+1)$ where $p_1 p_2 \cdots p_k$ is the factorization of $n$ into prime factors (not necessarily all distinct).

2004 Regional Olympiad - Republic of Srpska, 1
Tags: number theory proposed , number theory
Prove that the cube of any positive integer greater than 1 can be represented as a difference of the squares of two positive integers.

2000 Italy TST, 1
Tags: number theory proposed , number theory
Determine all triples $(x,y,z)$ of positive integers such that \[\frac{13}{x^2}+\frac{1996}{y^2}=\frac{z}{1997} \]

2012 Iran MO (3rd Round), 4
Tags: algebra , polynomial , Ring Theory , modular arithmetic , number theory proposed , number theory
$P(x)$ and $Q(x)$ are two polynomials with integer coefficients such that $P(x)|Q(x)^2+1$. [b]a)[/b] Prove that there exists polynomials $A(x)$ and $B(x)$ with rational coefficients and a rational number $c$ such that $P(x)=c(A(x)^2+B(x)^2)$. [b]b)[/b] If $P(x)$ is a monic polynomial with integer coefficients, Prove that there exists two polynomials $A(x)$ and $B(x)$ with integer coefficients such that $P(x)$ can be written in the form of $A(x)^2+B(x)^2$. [i]Proposed by Mohammad Gharakhani[/i]

1998 Baltic Way, 2
Tags: number theory proposed , number theory
A triple $(a,b,c)$ of positive integers is called [i]quasi-Pythagorean[/i] if there exists a triangle with lengths of the sides $a,b,c$ and the angle opposite to the side $c$ equal to $120^{\circ}$. Prove that if $(a,b,c)$ is a quasi-Pythagorean triple then $c$ has a prime divisor bigger than $5$.

1995 Irish Math Olympiad, 2
Tags: Diophantine equation , number theory proposed , number theory
Determine all integers $ a$ for which the equation $ x^2\plus{}axy\plus{}y^2\equal{}1$ has infinitely many distinct integer solutions $ x,y$.

2009 Baltic Way, 3
Tags: algebra , polynomial , number theory proposed , number theory
Let $ n$ be a given positive integer. Show that we can choose numbers $ c_k\in\{\minus{}1,1\}$ ($ i\le k\le n$) such that \[ 0\le\sum_{k\equal{}1}^nc_k\cdot k^2\le4.\]

2004 Iran MO (3rd Round), 14
Tags: number theory proposed , number theory
We define $ f: \mathbb{N} \rightarrow \mathbb{N}$, $ f(n) \equal{} \sum_{k \equal{} 1}^{n}(k,n)$. a) Show that if $ \gcd(m,n)\equal{}1$ then we have $ f(mn)\equal{}f(m)\cdot f(n)$; b) Show that $ \sum_{d|n}f(d) \equal{} nd(n)$.

2007 Serbia National Math Olympiad, 3
Tags: modular arithmetic , inequalities , quadratics , number theory , greatest common divisor , number theory proposed
Determine all pairs of natural numbers $(x; n)$ that satisfy the equation \[x^{3}+2x+1 = 2^{n}.\]

2012 India National Olympiad, 2
Tags: number theory , prime numbers , divisibility tests , number theory proposed
Let $p_1<p_2<p_3<p_4$ and $q_1<q_2<q_3<q_4$ be two sets of prime numbers, such that $p_4 - p_1 = 8$ and $q_4 - q_1= 8$. Suppose $p_1 > 5$ and $q_1>5$. Prove that $30$ divides $p_1 - q_1$.

2003 Turkey Team Selection Test, 3
Tags: arithmetic sequence , number theory proposed , number theory
Is there an arithmetic sequence with a. $2003$ b. infinitely many terms such that each term is a power of a natural number with a degree greater than $1$?

1997 All-Russian Olympiad, 1
Tags: modular arithmetic , number theory proposed , number theory
Find all integer solutions of the equation $(x^2 - y^2)^2 = 1 + 16y$. [i]M. Sonkin[/i]

2000 JBMO ShortLists, 7
Tags: number theory proposed , number theory
Find all the pairs of positive integers $(m,n)$ such that the numbers $A=n^2+2mn+3m^2+3n$, $B=2n^2+3mn+m^2$, $C=3n^2+mn+2m^2$ are consecutive in some order.

2007 Baltic Way, 19
Tags: number theory proposed , number theory
Let $r$ and $k$ be positive integers such that all prime divisors of $r$ are greater than $50$. A positive integer, whose decimal representation (without leading zeroes) has at least $k$ digits, will be called [i]nice[/i] if every sequence of $k$ consecutive digits of this decimal representation forms a number (possibly with leading zeroes) which is a multiple of $r$. Prove that if there exist infinitely many nice numbers, then the number $10^k-1$ is nice as well.

2003 Iran MO (3rd Round), 19
Tags: number theory proposed , number theory
An integer $ n$ is called a good number if and only if $ |n|$ is not square of another intger. Find all integers $ m$ such that they can be written in infinitely many ways as sum of three different good numbers and product of these three numbers is square of an odd number.

2007 China Team Selection Test, 1
Tags: modular arithmetic , number theory proposed , number theory
Find all the pairs of positive integers $ (a,b)$ such that $ a^2 \plus{} b \minus{} 1$ is a power of prime number $ ; a^2 \plus{} b \plus{} 1$ can divide $ b^2 \minus{} a^3 \minus{} 1,$ but it can't divide $ (a \plus{} b \minus{} 1)^2.$

2009 Czech and Slovak Olympiad III A, 1
Tags: modular arithmetic , number theory proposed , number theory
Knowing that the numbers $p, 3p+2, 5p+4, 7p+6, 9p+8$, and $11p+10$ are all primes, prove that $6p+11$ is a composite number.

2006 Switzerland Team Selection Test, 2
Tags: number theory proposed , number theory
Find all naturals $k$ such that $3^k+5^k$ is the power of a natural number with the exponent $\ge 2$.

2002 Tournament Of Towns, 2
Tags: number theory proposed , number theory
John and Mary select a natural number each and tell that to Bill. Bill wrote their sum and product in two papers hid one paper and showed the other to John and Mary. John looked at the number (which was $2002$ ) and declared he couldn't determine Mary's number. Knowing this Mary also said she couldn't determine John's number as well. What was Mary's Number?

2013 Greece National Olympiad, 2
Tags: quadratics , number theory proposed , number theory , Greece , 2013 , #2 , Diophantine equation
Solve in integers the following equation: \[y=2x^2+5xy+3y^2\]

2007 Iran MO (3rd Round), 1
Tags: modular arithmetic , number theory proposed , number theory
Let $ n$ be a natural number, such that $ (n,2(2^{1386}\minus{}1))\equal{}1$. Let $ \{a_{1},a_{2},\dots,a_{\varphi(n)}\}$ be a reduced residue system for $ n$. Prove that:\[ n|a_{1}^{1386}\plus{}a_{2}^{1386}\plus{}\dots\plus{}a_{\varphi(n)}^{1386}\]

2013 Bosnia Herzegovina Team Selection Test, 4
Tags: inequalities , symmetry , number theory proposed , number theory
Find all primes $p,q$ such that $p$ divides $30q-1$ and $q$ divides $30p-1$.