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

2009 Stars Of Mathematics, 4
Tags: algebra , polynomial , number theory proposed , number theory
Determine all non-constant polynomials $ f\in \mathbb{Z}[X]$ with the property that there exists $ k\in\mathbb{N}^*$ such that for any prime number $ p$, $ f(p)$ has at most $ k$ distinct prime divisors.

2010 Iran MO (3rd Round), 8
Tags: induction , number theory proposed , number theory
[b]numbers $n^2+1$[/b] Prove that there are infinitely many natural numbers of the form $n^2+1$ such that they don't have any divisor of the form $k^2+1$ except $1$ and themselves. time allowed for this question was 45 minutes.

2010 Contests, 3
Tags: number theory proposed , number theory
A student adds up rational fractions incorrectly: \[\frac{a}{b}+\frac{x}{y}=\frac{a+x}{b+y}\quad (\star) \] Despite that, he sometimes obtains correct results. For a given fraction $\frac{a}{b},a,b\in\mathbb{Z},b>0$, find all fractions $\frac{x}{y},x,y\in\mathbb{Z},y>0$ such that the result obtained by $(\star)$ is correct.

2004 Bulgaria Team Selection Test, 2
Tags: floor function , number theory proposed , number theory
Find all primes $p \ge 3$ such that $p- \lfloor p/q \rfloor q$ is a square-free integer for any prime $q<p$.

1978 Miklós Schweitzer, 3
Tags: logarithms , number theory proposed , number theory , college contests , combinatorics
Let $ 1<a_1<a_2< \ldots <a_n<x$ be positive integers such that $ \sum_{i\equal{}1}^n 1/a_i \leq 1$. Let $ y$ denote the number of positive integers smaller that $ x$ not divisible by any of the $ a_i$. Prove that \[ y > \frac{cx}{\log x}\] with a suitable positive constant $ c$ (independent of $ x$ and the numbers $ a_i$). [i]I. Z. Ruzsa[/i]

2013 Iran MO (2nd Round), 1
Tags: number theory proposed , number theory
Find all pairs $(a,b)$ of positive integers for which $\gcd(a,b)=1$, and $\frac{a}{b}=\overline{b.a}$. (For example, if $a=92$ and $b=13$, then $b/a=13.92$ )

2010 Portugal MO, 3
Tags: analytic geometry , modular arithmetic , number theory proposed , number theory
Consider a square $(p-1)\times(p-1)$, where $p$ is a prime number, which is divided by squares $1\times 1$ whose sides are parallel to the initial square's sides. Show that it is possible to select $p$ vertices such that there are no three collinear vertices.

2007 Baltic Way, 18
Tags: number theory proposed , number theory
Let $a,b,c,d$ be non-zero integers, such that the only quadruple of integers $(x, y, z, t)$ satisfying the equation \[ax^2+by^2+cz^2+dt^2=0\] is $x=y=z=t=0$. Does it follow that the numbers $a,b,c,d$ have the same sign?

1986 IMO Longlists, 53
Tags: number theory proposed , number theory
For given positive integers $r, v, n$ let $S(r, v, n)$ denote the number of $n$-tuples of non-negative integers $(x_1, \cdots, x_n)$ satisfying the equation $x_1 +\cdots+ x_n = r$ and such that $x_i \leq v$ for $i = 1, \cdots , n$. Prove that \[S(r, v, n)=\sum_{k=0}^{m} (-1)^k \binom nk \binom{r - (v + 1)k + n - 1}{n-1}\] Where $m=\left\{n,\left[\frac{r}{v+1}\right]\right\}.$

2005 Italy TST, 3
Tags: function , number theory proposed , number theory
The function $\psi : \mathbb{N}\rightarrow\mathbb{N}$ is defined by $\psi (n)=\sum_{k=1}^n\gcd (k,n)$. $(a)$ Prove that $\psi (mn)=\psi (m)\psi (n)$ for every two coprime $m,n \in \mathbb{N}$. $(b)$ Prove that for each $a\in\mathbb{N}$ the equation $\psi (x)=ax$ has a solution.

2003 Bundeswettbewerb Mathematik, 1
Tags: number theory proposed , number theory
Given six consecutive positive integers, prove that there exists a prime such that one and only one of these six integers is divisible by this prime.

2025 Bundeswettbewerb Mathematik, 1
Tags: number theory , number theory proposed
Fridolin the frog jumps on the number line: He starts at $0$, then jumps in some order on each of the numbers $1,2,\dots,9$ exactly once and finally returns with his last jump to $0$. Can the total distance he travelled with these $10$ jumps be a) $20$, b) $25$?

2011 Singapore MO Open, 5
Tags: modular arithmetic , algebra , function , domain , number theory proposed , number theory
Find all pairs of positive integers $(m,n)$ such that \[m+n-\frac{3mn}{m+n}=\frac{2011}{3}.\]

1995 Baltic Way, 2
Tags: number theory proposed , number theory
Let $a$ and $k$ be positive integers such that $a^2+k$ divides $(a-1)a(a+1)$. Prove that $k\ge a$.

2009 China Team Selection Test, 1
Tags: inequalities , number theory proposed , number theory
Let $ a > b > 1, b$ is an odd number, let $ n$ be a positive integer. If $ b^n|a^n\minus{}1,$ then $ a^b > \frac {3^n}{n}.$

2001 Baltic Way, 16
Tags: function , induction , number theory proposed , number theory
Let $f$ be a real-valued function defined on the positive integers satisfying the following condition: For all $n>1$ there exists a prime divisor $p$ of $n$ such that $f(n)=f\left(\frac{n}{p}\right)-f(p)$. Given that $f(2001)=1$, what is the value of $f(2002)$?

2013 ELMO Shortlist, 7
Tags: algebra , polynomial , modular arithmetic , number theory proposed , number theory
Let $p$ be a prime satisfying $p^2\mid 2^{p-1}-1$, and let $n$ be a positive integer. Define \[ f(x) = \frac{(x-1)^{p^n}-(x^{p^n}-1)}{p(x-1)}. \] Find the largest positive integer $N$ such that there exist polynomials $g(x)$, $h(x)$ with integer coefficients and an integer $r$ satisfying $f(x) = (x-r)^N g(x) + p \cdot h(x)$. [i]Proposed by Victor Wang[/i]

2008 Romania Team Selection Test, 2
Tags: modular arithmetic , number theory proposed , number theory
Let $ m, n \geq 1$ be two coprime integers and let also $ s$ an arbitrary integer. Determine the number of subsets $ A$ of $ \{1, 2, ..., m \plus{} n \minus{} 1\}$ such that $ |A| \equal{} m$ and $ \sum_{x \in A} x \equiv s \pmod{n}$.

2013 IFYM, Sozopol, 4
Tags: algebra , polynomial , number theory , greatest common divisor , modular arithmetic , number theory proposed , nice problem
Find all pairs of integers $(m,n)$ such that $m^6 = n^{n+1} + n -1$.

2001 India IMO Training Camp, 2
Tags: modular arithmetic , number theory proposed , number theory
Let $p > 3$ be a prime. For each $k\in \{1,2, \ldots , p-1\}$, define $x_k$ to be the unique integer in $\{1, \ldots, p-1\}$ such that $kx_k\equiv 1 \pmod{p}$ and set $kx_k = 1+ pn_k$. Prove that : \[\sum_{k=1}^{p-1}kn_k \equiv \frac{p-1}{2} \pmod{p}\]

2014 Iran Team Selection Test, 2
Tags: function , number theory , prime numbers , number theory proposed
is there a function $f:\mathbb{N}\rightarrow \mathbb{N}$ such that $i) \exists n\in \mathbb{N}:f(n)\neq n$ $ii)$ the number of divisors of $m$ is $f(n)$ if and only if the number of divisors of $f(m)$ is $n$

1996 Romania Team Selection Test, 2
Tags: number theory proposed , number theory
Find the greatest positive integer $n$ for which there exist $n$ nonnegative integers $x_1, x_2,\ldots , x_n$, not all zero, such that for any $\varepsilon_1, \varepsilon_2, \ldots, \varepsilon_n$ from the set $\{-1, 0, 1\}$, not all zero, $\varepsilon_1 x_1 + \varepsilon_2 x_2 + \cdots + \varepsilon_n x_n$ is not divisible by $n^3$.

2012 ELMO Shortlist, 1
Tags: modular arithmetic , quadratics , algorithm , number theory , number theory proposed
Find all positive integers $n$ such that $4^n+6^n+9^n$ is a square. [i]David Yang, Alex Zhu.[/i]

2006 Indonesia MO, 2
Tags: algebra , polynomial , number theory proposed , number theory
Let $ a,b,c$ be positive integers. If $ 30|a\plus{}b\plus{}c$, prove that $ 30|a^5\plus{}b^5\plus{}c^5$.

Oliforum Contest IV 2013, 1
Tags: modular arithmetic , number theory , greatest common divisor , number theory proposed , Contest 8
Given a prime $p$, consider integers $0<a<b<c<d<p$ such that $a^4\equiv b^4\equiv c^4\equiv d^4\pmod{p}$. Show that \[a+b+c+d\mid a^{2013}+b^{2013}+c^{2013}+d^{2013}\]