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

Estonia Open Junior - geometry, 2007.1.4
Tags: calculus , integration , ratio , geometry , perimeter , number theory unsolved , number theory
Call a scalene triangle K [i]disguisable[/i] if there exists a triangle K′ similar to K with two shorter sides precisely as long as the two longer sides of K, respectively. Call a disguisable triangle [i]integral[/i] if the lengths of all its sides are integers. (a) Find the side lengths of the integral disguisable triangle with the smallest possible perimeter. (b) Let K be an arbitrary integral disguisable triangle for which no smaller integral disguisable triangle similar to it exists. Prove that at least two side lengths of K are perfect squares.

2013 Dutch IMO TST, 3
Tags: number theory , prime numbers , number theory unsolved
Fix a sequence $a_1,a_2,a_3\ldots$ of integers satisfying the following condition:for all prime numbers $p$ and all positive integers $k$,we have $a_{pk+1}=pa_k-3a_p+13$.Determine all possible values of $a_{2013}$.

2006 China Team Selection Test, 3
Tags: number theory unsolved , number theory
For a positive integer $M$, if there exist integers $a$, $b$, $c$ and $d$ so that: \[ M \leq a < b \leq c < d \leq M+49, \qquad ad=bc \] then we call $M$ a GOOD number, if not then $M$ is BAD. Please find the greatest GOOD number and the smallest BAD number.

2004 Postal Coaching, 6
Tags: number theory unsolved , number theory
Find the number of ordered palindromic partitions of an integer $n$.

2008 Middle European Mathematical Olympiad, 4
Tags: number theory unsolved , number theory
Prove: If the sum of all positive divisors of $ n \in \mathbb{Z}^{\plus{}}$ is a power of two, then the number/amount of the divisors is a power of two.

2003 South africa National Olympiad, 3
Tags: number theory unsolved , number theory
The first four digits of a certain positive integer $n$ are $1137$. Prove that the digits of $n$ can be shuffled in such a way that the new number is divisible by 7.

2005 India National Olympiad, 2
Tags: number theory unsolved , number theory
Let $\alpha$ and $\beta$ be positive integers such that $\dfrac{43}{197} < \dfrac{ \alpha }{ \beta } < \dfrac{17}{77}$. Find the minimum possible value of $\beta$.

1979 IMO Longlists, 45
Tags: modular arithmetic , function , number theory unsolved , number theory
For any positive integer $n$, we denote by $F(n)$ the number of ways in which $n$ can be expressed as the sum of three different positive integers, without regard to order. Thus, since $10 = 7+2+1 = 6+3+1 = 5+4+1 = 5+3+2$, we have $F(10) = 4$. Show that $F(n)$ is even if $n \equiv 2$ or $4 \pmod 6$, but odd if $n$ is divisible by $6$.

2008 Indonesia TST, 4
Tags: number theory , greatest common divisor , number theory unsolved
Let $ a $ and $ b $ be natural numbers with property $ gcd(a,b)=1 $ . Find the least natural number $ k $ such that for every natural number $ r \ge k $ , there exist natural numbers $ m,n >1 $ in such a way that the number $ m^a n^b $ has exactly $ r+1 $ positive divisors.

1998 Vietnam Team Selection Test, 1
Tags: algebra , polynomial , calculus , integration , Rational Root Theorem , number theory unsolved , number theory
Find all integer polynomials $P(x)$, the highest coefficent is 1 such that: there exist infinitely irrational numbers $a$ such that $p(a)$ is a positive integer.

2008 Bundeswettbewerb Mathematik, 2
Tags: inequalities , number theory , greatest common divisor , least common multiple , relatively prime , number theory unsolved
Let the positive integers $ a,b,c$ chosen such that the quotients $ \frac{bc}{b\plus{}c},$ $ \frac{ca}{c\plus{}a}$ and $ \frac{ab}{a\plus{}b}$ are integers. Prove that $ a,b,c$ have a common divisor greater than 1.

2007 Nordic, 1
Tags: Diophantine equation , number theory unsolved , number theory
Find a solution to the equation $x^2-2x-2007y^2=0$ in positive integers.

2010 Contests, 2
Tags: number theory unsolved , number theory
Fifteen pairwise coprime positive integers chosen so that each of them less than 2010. Show that at least one of them is prime.

2007 China Girls Math Olympiad, 1
Tags: number theory , prime factorization , number theory unsolved
A positive integer $ m$ is called [i]good[/i] if there is a positive integer $ n$ such that $ m$ is the quotient of $ n$ by the number of positive integer divisors of $ n$ (including $ 1$ and $ n$ itself). Prove that $ 1, 2, \ldots, 17$ are good numbers and that $ 18$ is not a good number.

2009 Costa Rica - Final Round, 4
Tags: number theory unsolved , number theory
Show that the number $ 3^{{4}^{5}} \plus{} 4^{{5}^{6}}$ can be expresed as the product of two integers greater than $ 10^{2009}$

2001 India IMO Training Camp, 2
Tags: algebra , polynomial , number theory unsolved , number theory
Let $Q(x)$ be a cubic polynomial with integer coefficients. Suppose that a prime $p$ divides $Q(x_j)$ for $j = 1$ ,$2$ ,$3$ ,$4$ , where $x_1 , x_2 , x_3 , x_4$ are distinct integers from the set $\{0,1,\cdots, p-1\}$. Prove that $p$ divides all the coefficients of $Q(x)$.

2005 South East Mathematical Olympiad, 4
Tags: function , modular arithmetic , number theory unsolved , number theory
Find all positive integer solutions $(a, b, c)$ to the function $a^{2} + b^{2} + c^{2} = 2005$, where $a \leq b \leq c$.

2012 Federal Competition For Advanced Students, Part 2, 2
Tags: modular arithmetic , function , number theory unsolved , number theory
We define $N$ as the set of natural numbers $n<10^6$ with the following property: There exists an integer exponent $k$ with $1\le k \le 43$, such that $2012|n^k-1$. Find $|N|$.

1993 Turkey MO (2nd round), 6
Tags: number theory unsolved , number theory
$n_{1},\ldots ,n_{k}, a$ are integers that satisfies the above conditions A)For every $i\neq j$, $(n_{i}, n_{j})=1$ B)For every $i, a^{n_{i}}\equiv 1 (mod n_{i})$ C)For every $i, X^{a-1}\equiv 0(mod n_{i})$. Prove that $a^{x}\equiv 1(mod x)$ congruence has at least $2^{k+1}-2$ solutions. ($x>1$)

1979 IMO Longlists, 78
Tags: inequalities , number theory unsolved , number theory
Denote the number of different prime divisors of the number $n$ by $\omega (n)$, where $n$ is an integer greater than $1$. Prove that there exist infinitely many numbers $n$ for which $\omega (n)< \omega (n+1)<\omega (n+2)$ holds.

2006 China National Olympiad, 3
Tags: quadratics , number theory unsolved , number theory
Positive integers $k, m, n$ satisfy $mn=k^2+k+3$, prove that at least one of the equations $x^2+11y^2=4m$ and $x^2+11y^2=4n$ has an odd solution.

2012 India Regional Mathematical Olympiad, 6
Tags: induction , algebra , polynomial , number theory unsolved , number theory
Find all positive integers such that $3^{2n}+3n^2+7$ is a perfect square.

2007 Estonia National Olympiad, 1
Tags: quadratics , number theory unsolved , number theory
Find all real numbers a such that all solutions to the quadratic equation $ x^2 \minus{} ax \plus{} a \equal{} 0$ are integers.

2010 Tuymaada Olympiad, 4
Tags: floor function , algebra , polynomial , limit , number theory unsolved , number theory
Prove that for any positive real number $\alpha$, the number $\lfloor\alpha n^2\rfloor$ is even for infinitely many positive integers $n$.

2012 Baltic Way, 18
Tags: quadratics , number theory unsolved , number theory
Find all triples $(a,b,c)$ of integers satisfying $a^2 + b^2 + c^2 = 20122012$.