Found problems: 1766
1996 Iran MO (3rd Round), 1
Find all non-negative integer solutions of the equation
\[2^x + 3^y = z^2 .\]
2008 Junior Balkan Team Selection Tests - Romania, 1
Let $ p$ be a prime number, $ p\not \equal{} 3$, and integers $ a,b$ such that $p\mid a+b$ and $ p^2\mid a^3 \plus{} b^3$. Prove that $ p^2\mid a \plus{} b$ or $ p^3\mid a^3 \plus{} b^3$.
2006 Croatia Team Selection Test, 1
Find all natural numbers that can be expressed in a unique way as a sum of five or less perfect squares.
2006 Cono Sur Olympiad, 3
Let $n$ be a natural number. The finite sequence $\alpha$ of positive integer terms, there are $n$ different numbers ($\alpha$ can have repeated terms). Moreover, if from one from its terms any we subtract 1, we obtain a sequence which has, between its terms, at least $n$ different positive numbers. What's the minimum value of the sum of all the terms of $\alpha$?
2012 IberoAmerican, 3
Show that, for every positive integer $n$, there exist $n$ consecutive positive integers such that none is divisible by the sum of its digits.
(Alternative Formulation: Call a number good if it's not divisible by the sum of its digits. Show that for every positive integer $n$ there are $n$ consecutive good numbers.)
2003 Bulgaria Team Selection Test, 6
In natural numbers $m,n$ Solve : $n(n+1)(n+2)(n+3)=m(m+1)^2(m+2)^3(m+3)^4$
2013 China Team Selection Test, 1
Let $n\ge 2$ be an integer. $a_1,a_2,\dotsc,a_n$ are arbitrarily chosen positive integers with $(a_1,a_2,\dotsc,a_n)=1$. Let $A=a_1+a_2+\dotsb+a_n$ and $(A,a_i)=d_i$. Let $(a_2,a_3,\dotsc,a_n)=D_1, (a_1,a_3,\dotsc,a_n)=D_2,\dotsc, (a_1,a_2,\dotsc,a_{n-1})=D_n$.
Find the minimum of $\prod\limits_{i=1}^n\dfrac{A-a_i}{d_iD_i}$
2008 District Olympiad, 4
Let $ M$ be the set of those positive integers which are not divisible by $ 3$. The sum of $ 2n$ consecutive elements of $ M$ is $ 300$. Determine $ n$.
1987 Romania Team Selection Test, 2
Find all positive integers $A$ which can be represented in the form: \[ A = \left ( m - \dfrac 1n \right) \left( n - \dfrac 1p \right) \left( p - \dfrac 1m \right) \]
where $m\geq n\geq p \geq 1$ are integer numbers.
[i]Ioan Bogdan[/i]
2024 Bundeswettbewerb Mathematik, 1
Determine all pairs $(x,y)$ of integers satisfying
\[(x+2)^4-x^4=y^3.\]
2013 China National Olympiad, 2
For any positive integer $n$ and $0 \leqslant i \leqslant n$, denote $C_n^i \equiv c(n,i)\pmod{2}$, where $c(n,i) \in \left\{ {0,1} \right\}$. Define
\[f(n,q) = \sum\limits_{i = 0}^n {c(n,i){q^i}}\]
where $m,n,q$ are positive integers and $q + 1 \ne {2^\alpha }$ for any $\alpha \in \mathbb N$. Prove that if $f(m,q)\left| {f(n,q)} \right.$, then $f(m,r)\left| {f(n,r)} \right.$ for any positive integer $r$.
2012 ELMO Shortlist, 9
Are there positive integers $m,n$ such that there exist at least $2012$ positive integers $x$ such that both $m-x^2$ and $n-x^2$ are perfect squares?
[i]David Yang.[/i]
2018 Iran Team Selection Test, 5
Prove that for each positive integer $m$, one can find $m$ consecutive positive integers like $n$ such that the following phrase doesn't be a perfect power:
$$\left(1^3+2018^3\right)\left(2^3+2018^3\right)\cdots \left(n^3+2018^3\right)$$
[i]Proposed by Navid Safaei[/i]
2008 IberoAmerican Olympiad For University Students, 5
Find all positive integers $n$ such that there are positive integers $a_1,\cdots,a_n, b_1,\cdots,b_n$ that satisfy
\[(a_1^2+\cdots+a_n^2)(b_1^2+\cdots+b_n^2)-(a_1b_1+\cdots+a_nb_n)^2=n\]
2001 Romania National Olympiad, 1
Show that there exist no integers $a$ and $b$ such that $a^3+a^2b+ab^2+b^3=2001$.
2009 Junior Balkan Team Selection Test, 1
Find all two digit numbers $ \overline{AB}$ such that $ \overline{AB}$ divides $ \overline{A0B}$.
2008 Moldova Team Selection Test, 1
Determine a subset $ A\subset \mathbb{N}^*$ having $ 5$ different elements, so that the sum of the squares of its elements equals their product.
Do not simply post the subset, show how you found it.
2009 Spain Mathematical Olympiad, 1
Find all the finite sequences with $ n$ consecutive natural numbers $ a_1, a_2,..., a_n$, with $ n\geq3$ such that $ a_1\plus{} a_2\plus{}...\plus{} a_n\equal{}2009$.
1989 Federal Competition For Advanced Students, P2, 2
Find all triples $ (a,b,c)$ of integers with $ abc\equal{}1989$ and $ a\plus{}b\minus{}c\equal{}89$.
2010 Pan African, 1
a) Show that it is possible to pair off the numbers $1,2,3,\ldots ,10$ so that the sums of each of the five pairs are five different prime numbers.
b) Is it possible to pair off the numbers $1,2,3,\ldots ,20$ so that the sums of each of the ten pairs are ten different prime numbers?
2013 Greece Team Selection Test, 1
Find all pairs of non-negative integers $(m,n)$ satisfying $\frac{n(n+2)}{4}=m^4+m^2-m+1$
2007 Tournament Of Towns, 3
Determine all finite increasing arithmetic progressions in which each term is the reciprocal of a positive integer and the sum of all the terms is $1$.
2001 Iran MO (2nd round), 1
Let $n$ be a positive integer and $p$ be a prime number such that $np+1$ is a perfect square. Prove that $n+1$ can be written as the sum of $p$ perfect squares.
1993 Balkan MO, 4
Let $p$ be a prime and $m \geq 2$ be an integer. Prove that the equation \[ \frac{ x^p + y^p } 2 = \left( \frac{ x+y } 2 \right)^m \] has a positive integer solution $(x, y) \neq (1, 1)$ if and only if $m = p$.
[i]Romania[/i]
1979 IMO Longlists, 24
Let $a$ and $b$ be coprime integers, greater than or equal to $1$. Prove that all integers $n$ greater than or equal to $(a - 1)(b - 1)$ can be written in the form:
\[n = ua + vb, \qquad \text{with} (u, v) \in \mathbb N \times \mathbb N.\]