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

2006 Iran MO (3rd Round), 3
Tags: abstract algebra , group theory , calculus , integration , invariant , number theory proposed , number theory
$L$ is a fullrank lattice in $\mathbb R^{2}$ and $K$ is a sub-lattice of $L$, that $\frac{A(K)}{A(L)}=m$. If $m$ is the least number that for each $x\in L$, $mx$ is in $K$. Prove that there exists a basis $\{x_{1},x_{2}\}$ for $L$ that $\{x_{1},mx_{2}\}$ is a basis for $K$.

2007 Pre-Preparation Course Examination, 3
Tags: number theory proposed , number theory
Prove that for each $ a\in\mathbb N$, there are infinitely many natural $ n$, such that \[ n\mid a^{n \minus{} a \plus{} 1} \minus{} 1. \]

2014 ISI Entrance Examination, 5
Tags: number theory proposed , number theory
Prove that sum of $12$ consecutive integers cannot be a square. Give an example of $11$ consecutive integers whose sum is a perfect square.

2010 Contests, 1
Tags: modular arithmetic , number theory proposed , number theory
Prove that the number of ordered triples $(x, y, z)$ such that $(x+y+z)^2 \equiv axyz \mod{p}$, where $gcd(a, p) = 1$ and $p$ is prime is $p^2 + 1$.

2010 Middle European Mathematical Olympiad, 11
Tags: geometry , 3D geometry , number theory proposed , number theory
For a nonnegative integer $n$, define $a_n$ to be the positive integer with decimal representation \[1\underbrace{0\ldots0}_{n}2\underbrace{0\ldots0}_{n}2\underbrace{0\ldots0}_{n}1\mbox{.}\] Prove that $\frac{a_n}{3}$ is always the sum of two positive perfect cubes but never the sum of two perfect squares. [i](4th Middle European Mathematical Olympiad, Team Competition, Problem 7)[/i]

2006 India IMO Training Camp, 2
Tags: number theory proposed , number theory
the positive divisors $d_1,d_2,\cdots,d_k$ of a positive integer $n$ are ordered \[1=d_1<d_2<\cdots<d_k=n\] Suppose $d_7^2+d_{15}^2=d_{16}^2$. Find all possible values of $d_{17}$.

2006 Iran MO (3rd Round), 3
Tags: geometry , geometric transformation , number theory proposed , number theory
For $A\subset\mathbb Z$ and $a,b\in\mathbb Z$. We define $aA+b: =\{ax+b|x\in A\}$. If $a\neq0$ then we calll $aA+b$ and $A$ to similar sets. In this question the Cantor set $C$ is the number of non-negative integers that in their base-3 representation there is no $1$ digit. You see \[C=(3C)\dot\cup(3C+2)\ \ \ \ \ \ (1)\] (i.e. $C$ is partitioned to sets $3C$ and $3C+2$). We give another example $C=(3C)\dot\cup(9C+6)\dot\cup(3C+2)$. A representation of $C$ is a partition of $C$ to some similiar sets. i.e. \[C=\bigcup_{i=1}^{n}C_{i}\ \ \ \ \ \ (2)\] and $C_{i}=a_{i}C+b_{i}$ are similar to $C$. We call a representation of $C$ a primitive representation iff union of some of $C_{i}$ is not a set similar and not equal to $C$. Consider a primitive representation of Cantor set. Prove that a) $a_{i}>1$. b) $a_{i}$ are powers of 3. c) $a_{i}>b_{i}$ d) (1) is the only primitive representation of $C$.

2006 Junior Balkan Team Selection Tests - Moldova, 2
Tags: number theory proposed , number theory
Prove that there infinitely many numbers of the form $18^{m}+45^{m}+50^{m}+125^{m}$, divisible by 2006. $m\in N$

2014 All-Russian Olympiad, 1
Tags: number theory , prime numbers , number theory proposed
Call a natural number $n$ [i]good[/i] if for any natural divisor $a$ of $n$, we have that $a+1$ is also divisor of $n+1$. Find all good natural numbers. [i]S. Berlov[/i]

2010 Indonesia TST, 3
Tags: number theory proposed , number theory
Let $ x$, $ y$, and $ z$ be integers satisfying the equation \[ \dfrac{2008}{41y^2}\equal{}\dfrac{2z}{2009}\plus{}\dfrac{2007}{2x^2}.\] Determine the greatest value that $ z$ can take. [i]Budi Surodjo, Jogjakarta[/i]

2020 German National Olympiad, 4
Tags: number theory , number theory proposed , Divisibility , Divisors
Determine all positive integers $n$ for which there exists a positive integer $d$ with the property that $n$ is divisible by $d$ and $n^2+d^2$ is divisible by $d^2n+1$.

1995 All-Russian Olympiad, 5
Tags: algebra , difference of squares , special factorizations , number theory proposed , number theory
Prove that for every natural number $a_1>1$ there exists an increasing sequence of natural numbers $a_n$ such that $a^2_1+a^2_2+\cdots+a^2_k$ is divisible by $a_1+a_2+\cdots+a_k$ for all $k \geq 1$. [i]A. Golovanov[/i]

2014 Contests, 2
Tags: modular arithmetic , number theory proposed , number theory
Find all integers $n$, $n>1$, with the following property: for all $k$, $0\le k < n$, there exists a multiple of $n$ whose digits sum leaves a remainder of $k$ when divided by $n$.

2005 QEDMO 1st, 11 (Z3)
Tags: modular arithmetic , number theory proposed , number theory
Let $a,b,c$ be positive integers such that $a^2+b^2+c^2$ is divisble by $a+b+c$. Prove that at least two of the numbers $a^3,b^3,c^3$ leave the same remainder by division through $a+b+c$.

2013 Iran MO (3rd Round), 3
Tags: modular arithmetic , number theory proposed , number theory
Let $p>3$ a prime number. Prove that there exist $x,y \in \mathbb Z$ such that $p = 2x^2 + 3y^2$ if and only if $p \equiv 5, 11 \; (\mod 24)$ (20 points)

2008 Junior Balkan Team Selection Tests - Romania, 3
Tags: number theory proposed , number theory
Find all pairs $ (m,n)$ of integer numbers $ m,n > 1$ with property that $ mn \minus{} 1\mid n^3 \minus{} 1$.

2013 IFYM, Sozopol, 3
Tags: induction , number theory , relatively prime , number theory proposed
Let $\phi(n)$ be the number of positive integers less than $n$ that are relatively prime to $n$, where $n$ is a positive integer. Find all pairs of positive integers $(m,n)$ such that \[2^n + (n-\phi(n)-1)! = n^m+1.\]

2014 Contests, 2
Tags: modular arithmetic , number theory proposed , number theory
Find all all positive integers x,y,and z satisfying the equation $x^3=3^y7^z+8$

2010 Romania Team Selection Test, 3
Tags: modular arithmetic , arithmetic sequence , number theory proposed , number theory
Given a positive integer $a$, prove that $\sigma(am) < \sigma(am + 1)$ for infinitely many positive integers $m$. (Here $\sigma(n)$ is the sum of all positive divisors of the positive integer number $n$.) [i]Vlad Matei[/i]

2011 China Team Selection Test, 3
Tags: number theory proposed , number theory , combinatorics
A positive integer $n$ is known as an [i]interesting[/i] number if $n$ satisfies \[{\ \{\frac{n}{10^k}} \} > \frac{n}{10^{10}} \] for all $k=1,2,\ldots 9$. Find the number of interesting numbers.

1997 All-Russian Olympiad, 3
Tags: number theory proposed , number theory
Find all triples $m$; $n$; $l$ of natural numbers such that $m + n = gcd(m; n)^2$; $m + l = gcd(m; l)^2$; $n + l = gcd(n; l)^2$: [i]S. Tokarev[/i]

2011 Morocco TST, 2
Tags: number theory proposed , number theory
For positive integers $m$ and $n$, find the smalles possible value of $|2011^m-45^n|$. [i](Swiss Mathematical Olympiad, Final round, problem 3)[/i]

1998 Romania Team Selection Test, 3
Tags: number theory proposed , number theory
Let $m\ge 2$ be an integer. Find the smallest positive integer $n>m$ such that for any partition with two classes of the set $\{ m,m+1,\ldots ,n \}$ at least one of these classes contains three numbers $a,b,c$ (not necessarily different) such that $a^b=c$. [i]Ciprian Manolescu[/i]

2014 Argentina Cono Sur TST, 4
Tags: number theory proposed , number theory
Find all pairs of positive prime numbers $(p,q)$ such that $p^5+p^3+2=q^2-q$

1990 Turkey Team Selection Test, 6
Tags: abstract algebra , induction , number theory proposed , number theory
Let $k\geq 2$ and $n_1, \dots, n_k \in \mathbf{Z}^+$. If $n_2 | (2^{n_1} -1)$, $n_3 | (2^{n_2} -1)$, $\dots$, $n_k | (2^{n_{k-1}} -1)$, $n_1 | (2^{n_k} -1)$, show that $n_1 = \dots = n_k =1$.