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 Romania Team Selection Test, 1
Tags: quadratics , algebra , polynomial , number theory proposed , number theory
Find all triples $(a,b,c)$ of positive integers with the following property: for every prime $p$, if $n$ is a quadratic residue $\mod p$, then $an^2+bn+c$ is a quadratic residue $\mod p$.

2014 Mexico National Olympiad, 2
Tags: number theory proposed , number theory
A positive integer $a$ is said to [i]reduce[/i] to a positive integer $b$ if when dividing $a$ by its units digits the result is $b$. For example, 2015 reduces to $\frac{2015}{5} = 403$. Find all the positive integers that become 1 after some amount of reductions. For example, 12 is one such number because 12 reduces to 6 and 6 reduces to 1.

2012 JBMO TST - Turkey, 1
Tags: geometry , 3D geometry , number theory , least common multiple , number theory proposed , auyesl
Find the greatest positive integer $n$ for which $n$ is divisible by all positive integers whose cube is not greater than $n.$

2009 Benelux, 1
Tags: function , number theory proposed , number theory
Find all functions $f:\mathbb{Z}_{>0}\rightarrow\mathbb{Z}_{>0}$ that satisfy the following two conditions: [list]$\bullet\ f(n)$ is a perfect square for all $n\in\mathbb{Z}_{>0}$ $\bullet\ f(m+n)=f(m)+f(n)+2mn$ for all $m,n\in\mathbb{Z}_{>0}$.[/list]

2014 China Team Selection Test, 6
Tags: number theory proposed , number theory , poset
Let $k$ be a fixed even positive integer, $N$ is the product of $k$ distinct primes $p_1,...,p_k$, $a,b$ are two positive integers, $a,b\leq N$. Denote $S_1=\{d|$ $d|N, a\leq d\leq b, d$ has even number of prime factors$\}$, $S_2=\{d|$ $d|N, a\leq d\leq b, d$ has odd number of prime factors$\}$, Prove: $|S_1|-|S_2|\leq C^{\frac{k}{2}}_k$

2007 Iran MO (3rd Round), 2
Tags: graph theory , number theory proposed , number theory
Let $ m,n$ be two integers such that $ \varphi(m) \equal{}\varphi(n) \equal{} c$. Prove that there exist natural numbers $ b_{1},b_{2},\dots,b_{c}$ such that $ \{b_{1},b_{2},\dots,b_{c}\}$ is a reduced residue system with both $ m$ and $ n$.

2014 ELMO Shortlist, 9
Tags: logarithms , number theory proposed , number theory
Let $d$ be a positive integer and let $\varepsilon$ be any positive real. Prove that for all sufficiently large primes $p$ with $\gcd(p-1,d) \neq 1$, there exists an positive integer less than $p^r$ which is not a $d$th power modulo $p$, where $r$ is defined by \[ \log r = \varepsilon - \frac{1}{\gcd(d,p-1)}. \][i]Proposed by Shashwat Kishore[/i]

2006 Iran MO (3rd Round), 4
Tags: number theory proposed , number theory
$a,b,c,t$ are antural numbers and $k=c^{t}$ and $n=a^{k}-b^{k}$. a) Prove that if $k$ has at least $q$ different prime divisors, then $n$ has at least $qt$ different prime divisors. b)Prove that $\varphi(n)$ id divisible by $2^{\frac{t}{2}}$

2012 Korea National Olympiad, 1
Tags: induction , modular arithmetic , number theory proposed , number theory
$ p >3 $ is a prime number such that $ p | 2^{p-1} -1 $ and $ p \not | 2^x - 1 $ for $ x = 1, 2, \cdots , p-2 $. Let $ p = 2k+3 $. Now we define sequence $ \{ a_n \} $ as \[ a_i = a_{i+k}= 2^i ( 1 \le i \le k ) , \ a_{j+2k} = a_j a_{j+k} \ ( j \ge 1 ) \] Prove that there exist $2k$ consecutive terms of sequence $ a_{x+1} , a_{x+2} , \cdots , a_{x+2k} $ such that for all $ 1 \le i < j \le 2k $, $ a_{x+i} \not \equiv a_{x+j} \ (mod \ p) $.

2001 China National Olympiad, 2
Tags: number theory proposed , number theory
Let $X=\{1,2,\ldots,2001\}$. Find the least positive integer $m$ such that for each subset $W\subset X$ with $m$ elements, there exist $u,v\in W$ (not necessarily distinct) such that $u+v$ is of the form $2^{k}$, where $k$ is a positive integer.

1950 Miklós Schweitzer, 5
Tags: least common multiple , number theory proposed , number theory
Let $ 1\le a_1<a_2<\cdots<a_m\le N$ be a sequence of integers such that the least common multiple of any two of its elements is not greater than $ N$. Show that $ m\le 2\left[\sqrt{N}\right]$, where $ \left[\sqrt{N}\right]$ denotes the greatest integer $ \le \sqrt{N}$

2010 Contests, 1
Tags: number theory , prime numbers , number theory proposed
$a)$ Let $p$ and $q$ be distinct prime numbers such that $p+q^2$ divides $p^2+q$. Prove that $p+q^2$ divides $pq-1$. $b)$ Find all prime numbers $p$ such that $p+121$ divides $p^2+11$.

2008 Paraguay Mathematical Olympiad, 1
Tags: number theory , prime factorization , number theory proposed
How many positive integers $n < 500$ exist such that its prime factors are exclusively $2$, $7$, $11$, or a combination of these?

2014 Iran MO (3rd Round), 1
Tags: number theory proposed , number theory
Show that for every natural number $n$ there are $n$ natural numbers $ x_1 < x_2 < ... < x_n $ such that $$\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}-\frac{1}{x_1x_2...x_n}\in \mathbb{N}\cup {0}$$ (15 points )

1997 Iran MO (2nd round), 3
Tags: number theory proposed , number theory
Let $a,b$ be positive integers and $p=\frac{b}{4}\sqrt{\frac{2a-b}{2a+b}}$ be a prime number. Find the maximum value of $p$ and justify your answer.

2012 Iran Team Selection Test, 3
Tags: number theory proposed , number theory
Find all integer numbers $x$ and $y$ such that: \[(y^3+xy-1)(x^2+x-y)=(x^3-xy+1)(y^2+x-y).\] [i]Proposed by Mahyar Sefidgaran[/i]

2010 Baltic Way, 18
Tags: modular arithmetic , number theory proposed , number theory
Let $p$ be a prime number. For each $k$, $1\le k\le p-1$, there exists a unique integer denoted by $k^{-1}$ such that $1\le k^{-1}\le p-1$ and $k^{-1}\cdot k=1\pmod{p}$. Prove that the sequence \[1^{-1},\quad 1^{-1}+2^{-1},\quad 1^{-1}+2^{-1}+3^{-1},\quad \ldots ,\quad 1^{-1}+2^{-1}+\ldots +(p-1)^{-1} \] (addition modulo $p$) contains at most $\frac{p+1}{2}$ distinct elements.

2004 Polish MO Finals, 2
Tags: algebra , polynomial , search , number theory proposed , number theory
Let $ P$ be a polynomial with integer coefficients such that there are two distinct integers at which $ P$ takes coprime values. Show that there exists an infinite set of integers, such that the values $ P$ takes at them are pairwise coprime.

2010 Contests, 4
Tags: number theory , prime numbers , number theory proposed
Determine the smallest positive integer $m$ with the property that $m^3-3m^2+2m$ is divisible by both $79$ and $83$.

2014 Olympic Revenge, 4
Tags: limit , real analysis , number theory proposed , number theory , Analytic Number Theory
Let $a>1$ be a positive integer and $f\in \mathbb{Z}[x]$ with positive leading coefficient. Let $S$ be the set of integers $n$ such that \[n \mid a^{f(n)}-1.\] Prove that $S$ has density $0$; that is, prove that $\lim_{n\rightarrow \infty} \frac{|S\cap \{1,...,n\}|}{n}=0$.

2012 India IMO Training Camp, 2
Tags: number theory proposed , number theory
Find the least positive integer that cannot be represented as $\frac{2^a-2^b}{2^c-2^d}$ for some positive integers $a, b, c, d$.

1978 AMC 12/AHSME, 21
Tags: factorial , number theory proposed , number theory
$p$ and $q$ are distinct prime numbers. Prove that the number \[\frac {(pq-1)!} {p^{q-1}q^{p-1}(p-1)!(q-1)!}\] is an integer.

2010 IberoAmerican, 2
Tags: induction , number theory proposed , number theory
Determine if there are positive integers $a, b$ such that all terms of the sequence defined by \[ x_{1}= 2010,x_{2}= 2011\\ x_{n+2}= x_{n}+ x_{n+1}+a\sqrt{x_{n}x_{n+1}+b}\quad (n\ge 1) \] are integers.

2008 Iran MO (3rd Round), 5
Tags: algebra , polynomial , modular arithmetic , function , number theory proposed , number theory
Find all polynomials $ f\in\mathbb Z[x]$ such that for each $ a,b,x\in\mathbb N$ \[ a\plus{}b\plus{}c|f(a)\plus{}f(b)\plus{}f(c)\]

1982 Miklós Schweitzer, 4
Tags: logarithms , number theory proposed , number theory
Let \[ f(n)= \sum_{p|n , \;p^{\alpha} \leq n < p^{\alpha+1} \ } p^{\alpha} .\] Prove that \[ \limsup_{n \rightarrow \infty}f(n) \frac{ \log \log n}{n \log n}=1 .\] [i]P. Erdos[/i]