Found problems: 436
2012 QEDMO 11th, 1
Find all $x, y, z \in N_0$ with $(2^x + 1) (2^y-1) = 2^z-1$.
2019 Belarus Team Selection Test, 4.2
Four positive integers $x,y,z$ and $t$ satisfy the relations
\[ xy - zt = x + y = z + t. \]
Is it possible that both $xy$ and $zt$ are perfect squares?
1972 Swedish Mathematical Competition, 1
Find the largest real number $a$ such that \[\left\{ \begin{array}{l}
x - 4y = 1 \\
ax + 3y = 1\\
\end{array} \right.
\] has an integer solution.
1995 ITAMO, 6
Find all pairs of positive integers $x,y$ such that $x^2 +615 = 2^y$
2004 Thailand Mathematical Olympiad, 11
Find the number of positive integer solutions to $(x_1 + x_2 + x_3)(y_1 + y_2 + y_3 + y_4) = 91$
Oliforum Contest I 2008, 1
Let $ p>3$ be a prime. If $ p$ divides $ x$, prove that the equation $ x^2-1=y^p$ does not have positive integer solutions.
2004 Estonia National Olympiad, 1
Find all pairs of integers $(a, b)$ such that $a^2 + ab + b^2 = 1$
1993 All-Russian Olympiad Regional Round, 9.5
Show that the equation $x^3 +y^3 = 4(x^2y+xy^2 +1)$ has no integer solutions.
1996 Israel National Olympiad, 1
Let $a$ be a prime number and $n > 2$ an integer.
Find all integer solutions of the equation $x^n +ay^n = a^2z^n$
.
2012 NZMOC Camp Selection Problems, 6
Let $a, b$ and $c$ be positive integers such that $a^{b+c} = b^{c} c$. Prove that b is a divisor of $c$, and that $c$ is of the form $d^b$ for some positive integer $d$.
2013 Swedish Mathematical Competition, 1
For $r> 0$ denote by $B_r$ the set of points at distance at most $r$ length units from the origin.
If $P_r$ is the set of the points in $B_r$ whit integer coordinates, show that the equation $$xy^3z + 2x^3z^3-3x^5y = 0$$
has an odd number of solutions $(x, y, z)$ in $P_r$.
1905 Eotvos Mathematical Competition, 1
For given positive integers $n$ and $p$, find neaessary and sufficient conditions for the system of equations
$$x + py = n , \\ x + y = p^2$$
to have a solution $(x, y, z)$ of positive integers. Prove also that there is at most one such solution.
2014 Estonia Team Selection Test, 6
Find all natural numbers $n$ such that the equation $x^2 + y^2 + z^2 = nxyz$ has solutions in positive integers
2019 Romania National Olympiad, 3
Prove that the number of solutions in $ \left( \mathbb{N}\cup\{ 0 \} \right)\times \left( \mathbb{N}\cup\{ 0 \} \right)\times \left( \mathbb{N}\cup\{ 0 \} \right) $ of the parametric equation
$$ \sqrt{x^2+y+n}+\sqrt{y^2+x+n} = z, $$
is greater than zero and finite, for nay natural number $ n. $
2008 Denmark MO - Mohr Contest, 2
If three integers $p, q$ and $r$ apply that $$p + q^2 = r ^2.$$Show that $6$ adds up to $pqr$ .
1950 Poland - Second Round, 6
Solve the equation in integer numbers $$y^3-x^3=91$$
2016 India PRMO, 2
Find the number of integer solutions of the equation
$x^{2016} + (2016! + 1!) x^{2015} + (2015! + 2!) x^{2014} + ... + (1! + 2016!) = 0$
1998 Estonia National Olympiad, 4
Find all integers $n > 2$ for which $(2n)! = (n-2)!n!(n+2)!$ .
2021 New Zealand MO, 4
Find all triples $(x, p, n)$ of non-negative integers such that $p$ is prime and $2x(x + 5) = p^n + 3(x - 1)$.
1987 Dutch Mathematical Olympiad, 1
Solve into $N$: $$a^2 = 2^b +c^4$$
2014 Estonia Team Selection Test, 6
Find all natural numbers $n$ such that the equation $x^2 + y^2 + z^2 = nxyz$ has solutions in positive integers
2014 Junior Balkan Team Selection Tests - Romania, 2
Determine all pairs $(a, b)$ of integers which satisfy the equality $\frac{a + 2}{b + 1} +\frac{a + 1}{b + 2} = 1 +\frac{6}{a + b + 1}$
2019 Brazil EGMO TST, 1
We say that a triple of integers $(x, y, z)$ is of [i]jenifer [/i] type if $x, y$, and $z$ are positive integers, with $y \ge 2$, and $$x^2 - 3y^2 = z^2 - 3.$$
a) Find a triple $(x, y, z)$ of the jenifer type with $x = 5$ and $x = 7$.
b) Show that for every $x \ge 5$ and odd there are at least two distinct triples $(x, y_1, z_1)$ and $(x, y_2, z_2)$ of jenifer type.
c) Find some triple $(x, y, z)$ of jenifer type with $x$ even.
2009 Hanoi Open Mathematics Competitions, 8
Find all the pairs of the positive integers such that the product of the numbers of any pair plus the half of one of
the numbers plus one third of the other number is three times less than $1004$.
2020 Tournament Of Towns, 4
For some integer n the equation $x^2 + y^2 + z^2 -xy -yz - zx = n$ has an integer solution $x, y, z$. Prove that the equation$ x^2 + y^2 - xy = n$ also has an integer solution $x, y$.
Alexandr Yuran