Found problems: 1766
2010 Macedonia National Olympiad, 1
Solve the equation
\[ x^3+2y^3-4x-5y+z^2=2012, \]
in the set of integers.
1995 Baltic Way, 5
Let $a<b<c$ be three positive integers. Prove that among any $2c$ consecutive positive integers there exist three different numbers $x,y,z$ such that $abc$ divides $xyz$.
2012 Bosnia Herzegovina Team Selection Test, 4
Define a function $f:\mathbb{N}\rightarrow\mathbb{N}$, \[f(1)=p+1,\] \[f(n+1)=f(1)\cdot f(2)\cdots f(n)+p,\] where $p$ is a prime number. Find all $p$ such that there exists a natural number $k$ such that $f(k)$ is a perfect square.
2008 Bosnia Herzegovina Team Selection Test, 2
Find all pairs of positive integers $ m$ and $ n$ that satisfy (both) following conditions:
(i) $ m^{2}\minus{}n$ divides $ m\plus{}n^{2}$
(ii) $ n^{2}\minus{}m$ divides $ n\plus{}m^{2}$
2012 Indonesia MO, 1
Show that for any positive integers $a$ and $b$, the number \[n=\mathrm{LCM}(a,b)+\mathrm{GCD}(a,b)-a-b\] is an even non-negative integer.
[i]Proposer: Nanang Susyanto[/i]
2005 Baltic Way, 19
Is it possible to find $2005$ different positive square numbers such that their sum is also a square number ?
2008 Argentina Iberoamerican TST, 1
Find all integers $ x$ such that $ x(x\plus{}1)(x\plus{}7)(x\plus{}8)$ is a perfect square
It's a nice problem ...hope you enjoy it!
Daniel
2009 Benelux, 2
Let $n$ be a positive integer and let $k$ be an odd positive integer. Moreover, let $a,b$ and $c$ be integers (not necessarily positive) satisfying the equations
\[a^n+kb=b^n+kc=c^n+ka \]
Prove that $a=b=c$.
2010 Baltic Way, 16
For a positive integer $k$, let $d(k)$ denote the number of divisors of $k$ and let $s(k)$ denote the digit sum of $k$. A positive integer $n$ is said to be [i]amusing[/i] if there exists a positive integer $k$ such that $d(k)=s(k)=n$. What is the smallest amusing odd integer greater than $1$?
2014 Bulgaria National Olympiad, 1
Find all pairs of prime numbers $p\,,q$ for which:
\[p^2 \mid q^3 + 1 \,\,\, \text{and} \,\,\, q^2 \mid p^6-1\]
[i]Proposed by P. Boyvalenkov[/i]
2008 Silk Road, 1
Suppose $ a,c,d \in N$ and $ d|a^2b\plus{}c$ and $ d\geq a\plus{}c$
Prove that $ d\geq a\plus{}\sqrt[2b] {a}$
2009 China Team Selection Test, 1
Let $ n$ be a composite. Prove that there exists positive integer $ m$ satisfying $ m|n, m\le\sqrt {n},$ and $ d(n)\le d^3(m).$ Where $ d(k)$ denotes the number of positive divisors of positive integer $ k.$
2008 Bosnia And Herzegovina - Regional Olympiad, 2
Find all positive integers $ a$ and $ b$ such that $ \frac{a^{4}\plus{}a^{3}\plus{}1}{a^{2}b^{2}\plus{}ab^{2}\plus{}1}$ is an integer.
2010 ELMO Shortlist, 5
Find the set $S$ of primes such that $p \in S$ if and only if there exists an integer $x$ such that $x^{2010} + x^{2009} + \cdots + 1 \equiv p^{2010} \pmod{p^{2011}}$.
[i]Brian Hamrick.[/i]
2000 Flanders Math Olympiad, 1
An integer consists of 7 different digits, and is a multiple of each of its digits.
What digits are in this nubmer?
2024 Baltic Way, 16
Determine all composite positive integers $n$ such that, for each positive divisor $d$ of $n$, there are integers $k\geq 0$ and $m\geq 2$ such that $d=k^m+1$.
2006 Germany Team Selection Test, 1
Does there exist a natural number $n$ in whose decimal representation each digit occurs at least $2006$ times and which has the property that you can find two different digits in its decimal representation such that the number obtained from $n$ by interchanging these two digits is different from $n$ and has the same set of prime divisors as $n$ ?
2014 Contests, 1
Find all non-negative integer numbers $n$ for which there exists integers $a$ and $b$ such that $n^2=a+b$ and $n^3=a^2+b^2.$
2002 Romania Team Selection Test, 3
Let $n$ be a positive integer. $S$ is the set of nonnegative integers $a$ such that $1<a<n$ and $a^{a-1}-1$ is divisible by $n$. Prove that if $S=\{ n-1 \}$ then $n=2p$ where $p$ is a prime number.
[i]Mihai Cipu and Nicolae Ciprian Bonciocat[/i]
2007 Romania Team Selection Test, 4
i) Find all infinite arithmetic progressions formed with positive integers such that there exists a number $N \in \mathbb{N}$, such that for any prime $p$, $p > N$,
the $p$-th term of the progression is also prime.
ii) Find all polynomials $f(X) \in \mathbb{Z}[X]$, such that there exist $N \in \mathbb{N}$, such that for any prime $p$, $p > N$, $| f(p) |$ is also prime.
[i]Dan Schwarz[/i]
2013 India IMO Training Camp, 3
We define an operation $\oplus$ on the set $\{0, 1\}$ by
\[ 0 \oplus 0 = 0 \,, 0 \oplus 1 = 1 \,, 1 \oplus 0 = 1 \,, 1 \oplus 1 = 0 \,.\]
For two natural numbers $a$ and $b$, which are written in base $2$ as $a = (a_1a_2 \ldots a_k)_2$ and $b = (b_1b_2 \ldots b_k)_2$ (possibly with leading 0's), we define $a \oplus b = c$ where $c$ written in base $2$ is $(c_1c_2 \ldots c_k)_2$ with $c_i = a_i \oplus b_i$, for $1 \le i \le k$. For example, we have $7 \oplus 3 = 4$ since $ 7 = (111)_2$ and $3 = (011)_2$.
For a natural number $n$, let $f(n) = n \oplus \left[ n/2 \right]$, where $\left[ x \right]$ denotes the largest integer less than or equal to $x$. Prove that $f$ is a bijection on the set of natural numbers.
1999 Baltic Way, 16
Find the smallest positive integer $k$ which is representable in the form $k=19^n-5^m$ for some positive integers $m$ and $n$.
2013 Turkey Team Selection Test, 1
Let $\phi(n)$ be the number of positive integers less than $n$ that are relatively prime to $n$, where $n$ is a positive integer. Find all pairs of positive integers $(m,n)$ such that \[2^n + (n-\phi(n)-1)! = n^m+1.\]
2008 Mongolia Team Selection Test, 1
Given an integer $ a$. Let $ p$ is prime number such that $ p|a$ and $ p \equiv \pm 3 (mod8)$. Define a sequence $ \{a_n\}_{n \equal{} 0}^\infty$ such that $ a_n \equal{} 2^n \plus{} a$. Prove that the sequence $ \{a_n\}_{n \equal{} 0}^\infty$ has finitely number of square of integer.
2007 USAMO, 1
Let $n$ be a positive integer. Define a sequence by setting $a_{1}= n$ and, for each $k > 1$, letting $a_{k}$ be the unique integer in the range $0\leq a_{k}\leq k-1$ for which $a_{1}+a_{2}+...+a_{k}$ is divisible by $k$. For instance, when $n = 9$ the obtained sequence is $9,1,2,0,3,3,3,...$. Prove that for any $n$ the sequence $a_{1},a_{2},...$ eventually becomes constant.