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: 1362

2011 Kazakhstan National Olympiad, 3
Tags: number theory unsolved , number theory , Kazakhstan
Given are the odd integers $m> 1$, $k$, and a prime $p$ such that $p> mk +1$. Prove that $p^{2}\mid {\binom{k}{k}}^{m}+{\binom{k+1}{k}}^{m}+\cdots+{\binom{p-1}{k}}^{m}$.

2012 Mexico National Olympiad, 3
Tags: number theory unsolved , number theory
Prove among any $14$ consecutive positive integers there exist $6$ which are pairwise relatively prime.

1994 APMO, 5
Tags: floor function , logarithms , number theory unsolved , number theory
You are given three lists $A$, $B$, and $C$. List $A$ contains the numbers of the form $10^k$ in base $10$, with $k$ any integer greater than or equal to $1$. Lists $B$ and $C$ contain the same numbers translated into base $2$ and $5$ respectively: $$\begin{array}{lll} A & B & C \\ 10 & 1010 & 20 \\ 100 & 1100100 & 400 \\ 1000 & 1111101000 & 13000 \\ \vdots & \vdots & \vdots \end{array}$$ Prove that for every integer $n > 1$, there is exactly one number in exactly one of the lists $B$ or $C$ that has exactly $n$ digits.

2003 Indonesia MO, 1
Tags: number theory unsolved , number theory
Prove that $a^9 - a$ is divisible by $6$ for all integers $a$.

2001 All-Russian Olympiad, 1
Tags: ceiling function , floor function , number theory unsolved , number theory
The integers from $1$ to $999999$ are partitioned into two groups: the first group consists of those integers for which the closest perfect square is odd, whereas the second group consists of those for which the closest perfect square is even. In which group is the sum of the elements greater?

2007 Bundeswettbewerb Mathematik, 1
Tags: quadratics , induction , number theory unsolved , number theory
For which numbers $ n$ is there a positive integer $ k$ with the following property: The sum of digits for $ k$ is $ n$ and the number $ k^2$ has sum of digits $ n^2.$

2014 India IMO Training Camp, 2
Tags: number theory unsolved , number theory
Find all positive integers $x$ and $y$ such that $x^{x+y}=y^{3x}$.

1999 Brazil Team Selection Test, Problem 1
Tags: number theory unsolved , number theory
For a positive integer n, let $w(n)$ denote the number of distinct prime divisors of n. Determine the least positive integer k such that $2^{w(n)} \leq k \sqrt[4]{n}$ for all positive integers n.

2001 Brazil Team Selection Test, Problem 2
Tags: number theory unsolved , number theory
Let $f(n)$ denote the least positive integer $k$ such that $1+2+\cdots+k$ is divisible by $n$. Show that $f(n)=2n-1$ if and only if $n$ is a power of $2$.

2006 MOP Homework, 1
Tags: number theory unsolved , number theory
Let $n$ be an integer greater than $1$, and let $a_1$, $a_2$, ..., $a_n$ be not all identical positive integers. Prove that there are infinitely many primes $p$ such that $p$ divides $a_1^k+a_2^k+...+a_n^k$ for some positive integer $k$.

2004 China Team Selection Test, 1
Tags: number theory , prime factorization , number theory unsolved
Let $ m_1$, $ m_2$, $ \cdots$, $ m_r$ (may not distinct) and $ n_1$, $ n_2$ $ \cdots$, $ n_s$ (may not distinct) be two groups of positive integers such that for any positive integer $ d$ larger than $ 1$, the numbers of which can be divided by $ d$ in group $ m_1$, $ m_2$, $ \cdots$, $ m_r$ (including repeated numbers) are no less than that in group $ n_1$, $ n_2$ $ \cdots$, $ n_s$ (including repeated numbers). Prove that $ \displaystyle \frac{m_1 \cdot m_2 \cdots m_r}{n_1 \cdot n_2 \cdots n_s}$ is integer.

2009 Costa Rica - Final Round, 2
Tags: induction , number theory unsolved , number theory
Prove that for that for every positive integer $ n$, the smallest integer that is greater than $ (\sqrt {3} \plus{} 1)^{2n}$ is divisible by $ 2^{n \plus{} 1}$.

2008 Finnish National High School Mathematics Competition, 1
Tags: number theory unsolved , number theory
Foxes, wolves and bears arranged a big rabbit hunt. There were $45$ hunters catching $2008$ rabbits. Every fox caught $59$ rabbits, every wolf $41$ rabbits and every bear $40$ rabbits. How many foxes, wolves and bears were there in the hunting company?

2011 Tuymaada Olympiad, 4
Tags: geometry , 3D geometry , number theory unsolved , number theory
In a set of consecutive positive integers, there are exactly $100$ perfect cubes and $10$ perfect fourth powers. Prove that there are at least $2000$ perfect squares in the set.

2014 Singapore Senior Math Olympiad, 4
Tags: number theory unsolved , number theory
For each positive integer $n$ let \[x_n=p_1+\cdots+p_n\] where $p_1,\ldots,p_n$ are the first $n$ primes. Prove that for each positive integer $n$, there is an integer $k_n$ such that $x_n<k_n^2<x_{n+1}$

2014 Saudi Arabia BMO TST, 2
Tags: function , number theory unsolved , number theory
Let $\mathbb{N}$ denote the set of positive integers, and let $S$ be a set. There exists a function $f :\mathbb{N} \rightarrow S$ such that if $x$ and $y$ are a pair of positive integers with their difference being a prime number, then $f(x) \neq f(y)$. Determine the minimum number of elements in $S$.

2004 Vietnam National Olympiad, 3
Tags: calculus , integration , quadratics , modular arithmetic , number theory unsolved , number theory
Let $ S(n)$ be the sum of decimal digits of a natural number $ n$. Find the least value of $ S(m)$ if $ m$ is an integral multiple of $ 2003$.

1972 IMO Longlists, 1
Tags: number theory unsolved , number theory
Find all integer solutions of the equation \[1 + x + x^2 + x^3 + x^4 = y^4.\]

2004 Irish Math Olympiad, 1
Tags: number theory unsolved , number theory
1. (a) For which positive integers n, does 2n divide the sum of the first n positive integers? (b) Determine, with proof, those positive integers n (if any) which have the property that 2n + 1 divides the sum of the first n positive integers.

2009 Princeton University Math Competition, 3
Tags: number theory unsolved , number theory
Let $(x_n)$ be a sequence of positive integers defined as follows: $x_1$ is a fixed six-digit number and for any $n \geq 1$, $x_{n+1}$ is a prime divisor of $x_n + 1$. Find $x_{19} + x_{20}$.

2001 Tournament Of Towns, 2
Tags: number theory , least common multiple , induction , number theory unsolved
Do there exist positive integers $a_1<a_2<\ldots<a_{100}$ such that for $2\le k\le100$, the least common multiple of $a_{k-1}$ and $a_k$ is greater than the least common multiple of $a_k$ and $a_{k+1}$?

1982 USAMO, 4
Tags: AMC , USA(J)MO , USAMO , modular arithmetic , number theory unsolved , number theory
Prove that there exists a positive integer $k$ such that $k\cdot2^n+1$ is composite for every integer $n$.

2005 MOP Homework, 5
Tags: number theory unsolved , number theory
Find all integer solutions to $y^2(x^2+y^2-2xy-x-y)=(x+y)^2(x-y)$.

2005 Vietnam Team Selection Test, 2
Tags: abstract algebra , quadratics , number theory unsolved , number theory
Let $p\in \mathbb P,p>3$. Calcute: a)$S=\sum_{k=1}^{\frac{p-1}{2}} \left[\frac{2k^2}{p}\right]-2 \cdot \left[\frac{k^2}{p}\right]$ if $ p\equiv 1 \mod 4$ b) $T=\sum_{k=1}^{\frac{p-1}{2}} \left[\frac{k^2}{p}\right]$ if $p\equiv 1 \mod 8$

2009 Germany Team Selection Test, 1
Tags: number theory unsolved , number theory
Let $p > 7$ be a prime which leaves residue 1 when divided by 6. Let $m=2^p-1,$ then prove $2^{m-1}-1$ can be divided by $127m$ without residue.