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

2011 Iran MO (3rd Round), 5
Tags: modular arithmetic , number theory proposed , number theory
Suppose that $k$ is a natural number. Prove that there exists a prime number in $\mathbb Z_{[i]}$ such that every other prime number in $\mathbb Z_{[i]}$ has a distance at least $k$ with it.

2007 Korea - Final Round, 4
Tags: modular arithmetic , number theory , number theory proposed
Find all pairs $ (p, q)$ of primes such that $ {p}^{p}\plus{}{q}^{q}\plus{}1$ is divisible by $ pq$.

2002 Federal Competition For Advanced Students, Part 2, 2
Tags: number theory proposed , number theory
Let $b$ be a positive integer. Find all $2002$−tuples $(a_1, a_2,\ldots , a_{2002})$, of natural numbers such that \[\sum_{j=1}^{2002} a_j^{a_j}=2002b^b.\]

2011 Northern Summer Camp Of Mathematics, 2
Tags: function , number theory proposed , number theory
Find all functions $f: \mathbb N \cup \{0\} \to \mathbb N\cup \{0\}$ such that $f(1)>0$ and \[f(m^2+3n^2)=(f(m))^2 + 3(f(n))^2 \quad \forall m,n \in \mathbb N\cup \{0\}.\]

2008 Poland - Second Round, 1
Tags: number theory proposed , number theory
Determine the maximal possible length of the sequence of consecutive integers which are expressible in the form $ x^3\plus{}2y^2$, with $ x, y$ being integers.

2008 Romanian Master of Mathematics, 3
Tags: floor function , modular arithmetic , function , number theory , number theory proposed
Let $ a>1$ be a positive integer. Prove that every non-zero positive integer $ N$ has a multiple in the sequence $ (a_n)_{n\ge1}$, $ a_n\equal{}\left\lfloor\frac{a^n}n\right\rfloor$.

1998 Iran MO (3rd Round), 1
Tags: function , number theory proposed , number theory
Find all functions $f: \mathbb N \to \mathbb N$ such that for all positive integers $m,n$, [b](i)[/b] $mf(f(m))=\left( f(m) \right)^2$, [b](ii)[/b] If $\gcd(m,n)=d$, then $f(mn) \cdot f(d)=d \cdot f(m) \cdot f(n)$, [b](iii)[/b] $f(m)=m$ if and only if $m=1$.

2003 Hungary-Israel Binational, 3
Tags: induction , number theory proposed , number theory
Let $d > 0$ be an arbitrary real number. Consider the set $S_{n}(d)=\{s=\frac{1}{x_{1}}+\frac{1}{x_{2}}+...+\frac{1}{x_{n}}|x_{i}\in\mathbb{N},s<d\}$. Prove that $S_{n}(d)$ has a maximum element.

2006 Taiwan TST Round 1, 2
Tags: floor function , geometry , rectangle , polynomial , quadratics , number theory proposed , number theory
Let $p,q$ be two distinct odd primes. Calculate $\displaystyle \sum_{j=1}^{\frac{p-1}{2}}\left \lfloor \frac{qj}{p}\right \rfloor +\sum_{j=1}^{\frac{q-1}{2}}\left \lfloor \frac{pj}{q}\right\rfloor$.

2001 District Olympiad, 3
Tags: number theory proposed , number theory
Conside a positive odd integer $k$ and let $n_1<n_2<\ldots<n_k$ be $k$ positive odd integers. Prove that: \[n_1^2-n_2^2+n_3^2-n_4^2+\ldots+n_k^2\ge 2k^2-1\] [i]Titu Andreescu[/i]

1993 Korea - Final Round, 3
Tags: modular arithmetic , number theory , number theory proposed
Find the smallest $x \in\mathbb{N}$ for which $\frac{7x^{25}-10}{83}$ is an integer.

2012 Indonesia TST, 4
Tags: number theory , greatest common divisor , modular arithmetic , number theory proposed
Determine all integer $n > 1$ such that \[\gcd \left( n, \dfrac{n-m}{\gcd(n,m)} \right) = 1\] for all integer $1 \le m < n$.

2005 Turkey Team Selection Test, 1
Tags: search , function , number theory proposed , number theory
Show that for any integer $n\geq2$ and all integers $a_{1},a_{2},...,a_{n}$ the product $\prod_{i<j}{(a_{j}-a_{i})}$ is divisible by $\prod_{i<j}{(j-i)}$ .

2004 Romania Team Selection Test, 18
Tags: modular arithmetic , quadratics , number theory proposed , number theory
Let $p$ be a prime number and $f\in \mathbb{Z}[X]$ given by \[ f(x) = a_{p-1}x^{p-2} + a_{p-2}x^{p-3} + \cdots + a_2x+ a_1 , \] where $a_i = \left( \tfrac ip\right)$ is the Legendre symbol of $i$ with respect to $p$ (i.e. $a_i=1$ if $ i^{\frac {p-1}2} \equiv 1 \pmod p$ and $a_i=-1$ otherwise, for all $i=1,2,\ldots,p-1$). a) Prove that $f(x)$ is divisible with $(x-1)$, but not with $(x-1)^2$ iff $p \equiv 3 \pmod 4$; b) Prove that if $p\equiv 5 \pmod 8$ then $f(x)$ is divisible with $(x-1)^2$ but not with $(x-1)^3$. [i]Sugested by Calin Popescu.[/i]

2008 China Team Selection Test, 2
Tags: modular arithmetic , number theory , relatively prime , number theory proposed
Let $ n > 1$ be an integer, and $ n$ can divide $ 2^{\phi(n)} \plus{} 3^{\phi(n)} \plus{} \cdots \plus{} n^{\phi(n)},$ let $ p_{1},p_{2},\cdots,p_{k}$ be all distinct prime divisors of $ n$. Show that $ \frac {1}{p_{1}} \plus{} \frac {1}{p_{2}} \plus{} \cdots \plus{} \frac {1}{p_{k}} \plus{} \frac {1}{p_{1}p_{2}\cdots p_{k}}$ is an integer. ( where $ \phi(n)$ is defined as the number of positive integers $ \leq n$ that are relatively prime to $ n$.)

2001 Austrian-Polish Competition, 1
Tags: modular arithmetic , number theory proposed , number theory
Determine the number of positive integers $a$, so that there exist nonnegative integers $x_0,x_1,\ldots,x_{2001}$ which satisfy the equation \[ \displaystyle a^{x_0} = \sum_{i=1}^{2001} a^{x_i} \]

2009 Indonesia TST, 1
Tags: pigeonhole principle , search , number theory proposed , number theory
a. Does there exist 4 distinct positive integers such that the sum of any 3 of them is prime? b. Does there exist 5 distinct positive integers such that the sum of any 3 of them is prime?

2010 Contests, 1
Tags: number theory proposed , number theory
In a mathematics test number of participants is $N < 40$. The passmark is fixed at $65$. The test results are the following: The average of all participants is $66$, that of the promoted $71$ and that of the repeaters $56$. However, due to an error in the wording of a question, all scores are increased by $5$. At this point the average of the promoted participants becomes $75$ and that of the non-promoted $59$. (a) Find all possible values ​​of $N$. (b) Find all possible values ​​of $N$ in the case where, after the increase, the average of the promoted had become $79$ and that of non-promoted $47$.

2014 ELMO Shortlist, 2
Tags: modular arithmetic , number theory proposed , number theory
Define the Fibanocci sequence recursively by $F_1=1$, $F_2=1$ and $F_{i+2} = F_i + F_{i+1}$ for all $i$. Prove that for all integers $b,c>1$, there exists an integer $n$ such that the sum of the digits of $F_n$ when written in base $b$ is greater than $c$. [i]Proposed by Ryan Alweiss[/i]

2007 Moldova Team Selection Test, 2
Tags: modular arithmetic , algebra , polynomial , number theory , number theory proposed
Consider $p$ a prime number and $p$ consecutive positive integers $m_{1}, m_{2}, \ldots, m_{p}$. Choose a permutation $\sigma$ of $1, 2, \ldots, p$. Show that there exist two different numbers $k,l \in \{1,2, \ldots, p\}$ such that $m_{k}m_{\sigma(k)}-m_{l}m_{\sigma(l)}$ is divisible by $p$.

2012 Iran MO (3rd Round), 8
Tags: number theory proposed , number theory
[b]a)[/b] Does there exist an infinite subset $S$ of the natural numbers, such that $S\neq \mathbb{N}$, and such that for each natural number $n\not \in S$, exactly $n$ members of $S$ are coprime with $n$? [b]b)[/b] Does there exist an infinite subset $S$ of the natural numbers, such that for each natural number $n\in S$, exactly $n$ members of $S$ are coprime with $n$? [i]Proposed by Morteza Saghafian[/i]

2011 Bosnia Herzegovina Team Selection Test, 2
Tags: geometry , trigonometry , number theory , number theory proposed
On semicircle, with diameter $|AB|=d$, are given points $C$ and $D$ such that: $|BC|=|CD|=a$ and $|DA|=b$ where $a, b, d$ are different positive integers. Find minimum possible value of $d$

2007 IMAC Arhimede, 4
Tags: number theory , divides , Sum , Sums and Products , combinatorics , number theory proposed , pigeonhole principle
Prove that for any given number $a_k, 1 \le k \le 5$, there are $\lambda_k \in \{-1, 0, 1\}, 1 \le k \le 5$, which are not all equal zero, such that $11 | \lambda_1a_1^2+\lambda_2a_2^2+\lambda_3a_3^2+\lambda_4a_4^2+\lambda_5a_5^2$

2016 Middle European Mathematical Olympiad, 4
Tags: number theory , number theory proposed , functional equation , function , Divisibility , algebra
Find all $f : \mathbb{N} \to \mathbb{N} $ such that $f(a) + f(b)$ divides $2(a + b - 1)$ for all $a, b \in \mathbb{N}$. Remark: $\mathbb{N} = \{ 1, 2, 3, \ldots \} $ denotes the set of the positive integers.

2006 Greece National Olympiad, 2
Tags: modular arithmetic , number theory proposed , number theory
Let $n$ be a positive integer. Prove that the equation \[x+y+\frac{1}{x}+\frac{1}{y}=3n\] does not have solutions in positive rational numbers.