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

1978 IMO Longlists, 10
Tags: number theory , prime numbers , arithmetic sequence , number theory proposed
Show that for any natural number $n$ there exist two prime numbers $p$ and $q, p \neq q$, such that $n$ divides their difference.

2012 Paraguay Mathematical Olympiad, 1
Tags: number theory proposed , number theory
Define a list of number with the following properties: - The first number of the list is a one-digit natural number. - Each number (since the second) is obtained by adding $9$ to the number before in the list. - The number $2012$ is in that list. Find the first number of the list.

2010 Contests, 1
Tags: number theory , prime numbers , number theory proposed
a) Show that it is possible to pair off the numbers $1,2,3,\ldots ,10$ so that the sums of each of the five pairs are five different prime numbers. b) Is it possible to pair off the numbers $1,2,3,\ldots ,20$ so that the sums of each of the ten pairs are ten different prime numbers?

2014 Contests, 1
Tags: modular arithmetic , number theory , greatest common divisor , least common multiple , relatively prime , number theory proposed , Tuymaada
Four consecutive three-digit numbers are divided respectively by four consecutive two-digit numbers. What minimum number of different remainders can be obtained? [i](A. Golovanov)[/i]

2013 Greece Team Selection Test, 1
Tags: number theory proposed , number theory
Find all pairs of non-negative integers $(m,n)$ satisfying $\frac{n(n+2)}{4}=m^4+m^2-m+1$

2013 Iran Team Selection Test, 8
Tags: modular arithmetic , number theory , greatest common divisor , number theory proposed
Find all Arithmetic progressions $a_{1},a_{2},...$ of natural numbers for which there exists natural number $N>1$ such that for every $k\in \mathbb{N}$: $a_{1}a_{2}...a_{k}\mid a_{N+1}a_{N+2}...a_{N+k}$

1986 Iran MO (2nd round), 3
Tags: number theory proposed , number theory
Find the smallest positive integer for which when we move the last right digit of the number to the left, the remaining number be $\frac 32$ times of the original number.

2009 ITAMO, 3
Tags: number theory proposed , number theory
A natural number $n$ is called [i]nice[/i] if it enjoys the following properties: • The expression is made ​​up of $4$ decimal digits; • the first and third digits of $n$ are equal; • the second and fourth digits of $n$ are equal; • the product of the digits of $n$ divides $n^2$. Determine all nice numbers.

2012 International Zhautykov Olympiad, 1
Tags: function , induction , number theory proposed , number theory
Do there exist integers $m, n$ and a function $f\colon \mathbb R \to \mathbb R$ satisfying simultaneously the following two conditions? $\bullet$ i) $f(f(x))=2f(x)-x-2$ for any $x \in \mathbb R$; $\bullet$ ii) $m \leq n$ and $f(m)=n$.

2007 All-Russian Olympiad Regional Round, 11.5
Tags: number theory proposed , number theory
Find all positive integers $ n$ for which there exist integers $ a,b,c$ such that $ a\plus{}b\plus{}c\equal{}0$ and the number $ a^{n}\plus{}b^{n}\plus{}c^{n}$ is prime.

2012 Federal Competition For Advanced Students, Part 1, 2
Tags: modular arithmetic , number theory , number theory proposed
Determine all solutions $(n, k)$ of the equation $n!+An = n^k$ with $n, k \in\mathbb{N}$ for $A = 7$ and for $A = 2012$.

2002 France Team Selection Test, 2
Tags: number theory proposed , number theory
Consider the set $S$ of integers $k$ which are products of four distinct primes. Such an integer $k=p_1p_2p_3p_4$ has $16$ positive divisors $1=d_1<d_2<\ldots <d_{15}<d_{16}=k$. Find all elements of $S$ less than $2002$ such that $d_9-d_8=22$.

2009 Iran Team Selection Test, 11
Tags: modular arithmetic , number theory proposed , number theory
Let $n$ be a positive integer. Prove that \[ 3^{\dfrac{5^{2^n}-1}{2^{n+2}}} \equiv (-5)^{\dfrac{3^{2^n}-1}{2^{n+2}}} \pmod{2^{n+4}}. \]

1996 Irish Math Olympiad, 1
Tags: modular arithmetic , greatest common divisor , number theory , relatively prime , number theory proposed
For each positive integer $ n$, let $ f(n)$ denote the greatest common divisor of $ n!\plus{}1$ and $ (n\plus{}1)!$. Find, without proof, a formula for $ f(n)$.

1973 Miklós Schweitzer, 4
Tags: limit , logarithms , number theory proposed , number theory
Let $ f(n)$ be that largest integer $ k$ such that $ n^k$ divides $ n!$, and let $ F(n)\equal{} \max_{2 \leq m \leq n} f(m)$. Show that \[ \lim_{n\rightarrow \infty} \frac{F(n) \log n}{n \log \log n}\equal{}1.\] [i]P. Erdos[/i]

2013 Romania Team Selection Test, 3
Tags: quadratics , inequalities , induction , number theory proposed , number theory
Let $S$ be the set of all rational numbers expressible in the form \[\frac{(a_1^2+a_1-1)(a_2^2+a_2-1)\ldots (a_n^2+a_n-1)}{(b_1^2+b_1-1)(b_2^2+b_2-1)\ldots (b_n^2+b_n-1)}\] for some positive integers $n, a_1, a_2 ,\ldots, a_n, b_1, b_2, \ldots, b_n$. Prove that there is an infinite number of primes in $S$.

2005 QEDMO 1st, 1 (Z4)
Tags: geometry , 3D geometry , induction , modular arithmetic , number theory proposed , number theory
Prove that every integer can be written as sum of $5$ third powers of integers.

2003 Romania Team Selection Test, 18
Tags: induction , number theory proposed , number theory
For every positive integer $n$ we denote by $d(n)$ the sum of its digits in the decimal representation. Prove that for each positive integer $k$ there exists a positive integer $m$ such that the equation $x+d(x)=m$ has exactly $k$ solutions in the set of positive integers.

2011 Middle European Mathematical Olympiad, 1
Tags: quadratics , number theory proposed , number theory
Initially, only the integer $44$ is written on a board. An integer a on the board can be re- placed with four pairwise different integers $a_1, a_2, a_3, a_4$ such that the arithmetic mean $\frac 14 (a_1 + a_2 + a_3 + a_4)$ of the four new integers is equal to the number $a$. In a step we simultaneously replace all the integers on the board in the above way. After $30$ steps we end up with $n = 4^{30}$ integers $b_1, b2,\ldots, b_n$ on the board. Prove that \[\frac{b_1^2 + b_2^2+b_3^2+\cdots+b_n^2}{n}\geq 2011.\]

2010 Mexico National Olympiad, 1
Tags: number theory proposed , number theory
Find all triplets of natural numbers $(a,b,c)$ that satisfy the equation $abc=a+b+c+1$.

2013 IFYM, Sozopol, 6
Tags: number theory proposed , number theory
Prove that if $t$ is a natural number then there exists a natural number $n>1$ such that $(n,t)=1$ and none of the numbers $n+t,n^2+t,n^3+t,....$ are perfect powers.

2011 Indonesia MO, 6
Tags: modular arithmetic , number theory proposed , number theory
Let a sequence of integers $a_0, a_1, a_2, \cdots, a_{2010}$ such that $a_0 = 1$ and $2011$ divides $a_{k-1}a_k - k$ for all $k = 1, 2, \cdots, 2010$. Prove that $2011$ divides $a_{2010} + 1$.

2008 Serbia National Math Olympiad, 1
Tags: quadratics , number theory proposed , number theory
Find all nonegative integers $ x,y,z$ such that $ 12^x\plus{}y^4\equal{}2008^z$

2003 India IMO Training Camp, 10
Tags: modular arithmetic , number theory proposed , number theory
Let $n$ be a positive integer greater than $1$, and let $p$ be a prime such that $n$ divides $p-1$ and $p$ divides $n^3-1$. Prove that $4p-3$ is a square.

2005 Iran MO (3rd Round), 5
Tags: induction , number theory , prime numbers , number theory proposed
Let $a,b,c\in \mathbb N$ be such that $a,b\neq c$. Prove that there are infinitely many prime numbers $p$ for which there exists $n\in\mathbb N$ that $p|a^n+b^n-c^n$.