Found problems: 916
2001 Croatia Team Selection Test, 3
Find all solutions of the equation $(a^a)^5 = b^b$ in positive integers.
2021 Cyprus JBMO TST, 2
Find all pairs of natural numbers $(\alpha,\beta)$ for which, if $\delta$ is the greatest common divisor of $\alpha,\beta$, and $\varDelta$ is the least common multiple of $\alpha,\beta$, then
\[ \delta + \Delta = 4(\alpha + \beta) + 2021\]
2006 Federal Math Competition of S&M, Problem 2
Given prime numbers $p$ and $q$ with $p<q$, determine all pairs $(x,y)$ of positive integers such that
$$\frac1x+\frac1y=\frac1p-\frac1q.$$
2000 Kazakhstan National Olympiad, 4
Find all triples of natural numbers $ (x, y, z) $ that satisfy the condition $ (x + 1) ^ {y + 1} + 1 = (x + 2) ^ {z + 1}. $
2021 Israel TST, 4
Let $r$ be a positive integer and let $a_r$ be the number of solutions to the equation $3^x-2^y=r$ ,such that $0\leq x,y\leq 5781$ are integers. What is the maximal value of $a_r$?
PEN H Problems, 5
Find all pairs $(x, y)$ of rational numbers such that $y^2 =x^3 -3x+2$.
2014 IMAC Arhimede, 3
a) Prove that the equation $2^x + 21^x = y^3$ has no solution in the set of natural numbers.
b) Solve the equation $2^x + 21^y = z^2y$ in the set of non-negative integer numbers.
2011 Dutch IMO TST, 1
Find all pairs $(x, y)$ of integers that satisfy $x^2 + y^2 + 3^3 = 456\sqrt{x - y}$.
PEN H Problems, 8
Show that the equation \[x^{3}+y^{3}+z^{3}+t^{3}=1999\] has infinitely many integral solutions.
PEN H Problems, 33
Does there exist an integer such that its cube is equal to $3n^2 +3n+7$, where $n$ is integer?
2020 Austrian Junior Regional Competition, 4
Find all positive integers $a$ for which the equation $7an -3n! = 2020$ has a positive integer solution $n$.
(Richard Henner)
1995 India National Olympiad, 2
Show that there are infintely many pairs $(a,b)$ of relatively prime integers (not necessarily positive) such that both the equations \begin{eqnarray*} x^2 +ax +b &=& 0 \\ x^2 + 2ax + b &=& 0 \\ \end{eqnarray*} have integer roots.
2009 Croatia Team Selection Test, 4
Prove that there are infinite many positive integers $ n$ such that
$ n^2\plus{}1\mid n!$, and infinite many of those for which $ n^2\plus{}1 \nmid n!$.
1983 Brazil National Olympiad, 1
Show that there are only finitely many solutions to $1/a + 1/b + 1/c = 1/1983$ in positive integers.
1999 USAMTS Problems, 4
In $\triangle PQR$, $PQ=8$, $QR=13$, and $RP=15$. Prove that there is a point $S$ on line segment $\overline{PR}$, but not at its endpoints, such that $PS$ and $QS$ are also integers.
[asy]
size(200);
defaultpen(linewidth(0.8));
pair P=origin,Q=(8,0),R=(7,10),S=(3/2,15/7);
draw(P--Q--R--cycle);
label("$P$",P,W);
label("$Q$",Q,E);
label("$R$",R,NE);
draw(Q--S,linetype("4 4"));
label("$S$",S,NW);
[/asy]
1980 IMO, 3
Prove that the equation \[ x^n + 1 = y^{n+1}, \] where $n$ is a positive integer not smaller then 2, has no positive integer solutions in $x$ and $y$ for which $x$ and $n+1$ are relatively prime.
PEN H Problems, 15
Prove that there are no integers $x$ and $y$ satisfying $x^{2}=y^{5}-4$.
1967 IMO Shortlist, 3
Suppose $\tan \alpha = \dfrac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. Prove that the number $\tan \beta$ for which $\tan {2 \beta} = \tan {3 \alpha}$ is rational only when $p^2 + q^2$ is the square of an integer.
2024 Mozambique National Olympiad, P5
Find all pairs of positive integers $x,y$ such that $\frac{4}{x}+\frac{2}{y}=1$
2019 Thailand TSTST, 2
Find all nonnegative integers $x, y, z$ satisfying the equation $$2^x+31^y=z^2.$$
1995 Argentina National Olympiad, 2
For each positive integer $n$ let $p(n)$ be the number of ordered pairs $(x,y)$ of positive integers such that$$\dfrac{1}{x}+\dfrac{1}{y} =\dfrac{1}{n}.$$For example, for $n=2$ the pairs are $(3,6),(4,4),(6,3)$. Therefore $p(2)=3$.
a) Determine $p(n)$ for all $n$ and calculate $p(1995)$.
b) Determine all pairs $n$ such that $p(n)=3$.
2010 Contests, 2
Determine all triples $(x, y, z)$ of positive integers $x > y > z > 0$, such that $x^2 = y \cdot 2^z + 1$
2019 Chile National Olympiad, 3
Find all solutions $x,y,z$ in the positive integers of the equation $$3^x -5^y = z^2$$
2016 Postal Coaching, 2
Solve the equation for primes $p$ and $q$: $$p^3-q^3=pq^3-1.$$
2015 Czech and Slovak Olympiad III A, 1
Find all 4-digit numbers $n$, such that $n=pqr$, where $p<q<r$ are distinct primes, such that $p+q=r-q$ and $p+q+r=s^2$, where $s$ is a prime number.