Found problems: 916
2009 IMAC Arhimede, 5
Find all natural numbers $x$ and $y$ such that $x^y-y^x=1$ .
2019 Regional Olympiad of Mexico West, 5
Prove that for every integer $n > 1$ there exist integers $x$ and $y$ such that $$\frac{1}{n}=\frac{1}{x(x+1)}+\frac{1}{(x+1)(x+2)}+...+\frac{1}{y(y+1)}.$$
1969 IMO Shortlist, 7
$(BUL 1)$ Prove that the equation $\sqrt{x^3 + y^3 + z^3}=1969$ has no integral solutions.
2021 Kazakhstan National Olympiad, 5
Let $a$ be a positive integer. Prove that for any pair $(x,y)$ of integer solutions of equation $$x(y^2-2x^2)+x+y+a=0$$ we have: $$|x| \leqslant a+\sqrt{2a^2+2}$$
2010 Abels Math Contest (Norwegian MO) Final, 4a
Find all positive integers $k$ and $\ell$ such that $k^2 -\ell^2 = 1005$.
2011 Junior Balkan Team Selection Tests - Moldova, 6
Find the sum of the numbers written with two digits $\overline{ab}$ for which the equation $3^{x + y} =3^x + 3^y + \overline{ab}$ has at least one solution $(x, y)$ in natural numbers.
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]
2017 Thailand Mathematical Olympiad, 7
Show that no pairs of integers $(m, n)$ satisfy $2560m^2 + 5m + 6 = n^5$.
.
PEN H Problems, 7
Determine all pairs $(x,y)$ of positive integers satisfying the equation \[(x+y)^{2}-2(xy)^{2}=1.\]
2014 Cuba MO, 7
Find all pairs of integers $(a, b)$ that satisfy the equation
$$(a + 1)(b- 1) = a^2b^2.$$
2009 Bulgaria National Olympiad, 1
The natural numbers $a$ and $b$ satis fy the inequalities $a > b > 1$ . It is also known that the equation
$\frac{a^x - 1}{a - 1}=\frac{b^y - 1}{b - 1}$ has at least two solutions in natural numbers, when $x > 1$ and $y > 1$.
Prove that the numbers $a$ and $b$ are coprime (their greatest common divisor is $1$).
1998 Belarus Team Selection Test, 2
a) Given that integers $a$ and $b$ satisfy the equality $$a^2 - (b^2 - 4b + 1) a - (b^4 - 2b^3) = 0 \,\,\, (*)$$, prove that $b^2 + a$ is a square of an integer.
b) Do there exist an infinitely many of pairs $(a,b)$ satisfying (*)?
2015 Dutch IMO TST, 2
Determine all positive integers $n$ for which there exist positive integers $a_1,a_2, ..., a_n$
with $a_1 + 2a_2 + 3a_3 +... + na_n = 6n$ and $\frac{1}{a_1}+\frac{2}{a_2}+\frac{3}{a_3}+ ... +\frac{n}{a_n}= 2 + \frac1n$
2014 Purple Comet Problems, 11
Shenelle has some square tiles. Some of the tiles have side length $5\text{ cm}$ while the others have side length $3\text{ cm}$. The total area that can be covered by the tiles is exactly $2014\text{ cm}^2$. Find the least number of tiles that Shenelle can have.
2018 Finnish National High School Mathematics Comp, 5
Solve the diophantine equation $x^{2018}-y^{2018}=(xy)^{2017}$ when $x$ and $y$ are non-negative integers.
PEN H Problems, 26
Solve in integers the following equation \[n^{2002}=m(m+n)(m+2n)\cdots(m+2001n).\]
2018 Hanoi Open Mathematics Competitions, 6
Nam spent $20$ dollars for $20$ stationery items consisting of books, pens and pencils. Each book, pen, and pencil cost $3$ dollars, $1.5$ dollars and $0.5$ dollar respectively. How many dollars did Nam spend for books?
2022 3rd Memorial "Aleksandar Blazhevski-Cane", P3
Find all triplets of positive integers $(x, y, z)$ such that $x^2 + y^2 + x + y + z = xyz + 1$.
[i]Proposed by Viktor Simjanoski[/i]
2023 Brazil National Olympiad, 4
Determine the smallest integer $k$ for which there are three distinct positive integers $a$, $b$ and $c$, such that $$a^2 =bc \text{ and } k = 2b+3c-a.$$
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.
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.
2022 Dutch IMO TST, 1
Determine all positive integers $n \ge 2$ which have a positive divisor $m | n$ satisfying $$n = d^3 + m^3.$$
where $d$ is the smallest divisor of $n$ which is greater than $1$.
2012 International Zhautykov Olympiad, 3
Find all integer solutions of the equation the equation $2x^2-y^{14}=1$.
2015 BAMO, 3
Let $k$ be a positive integer. Prove that there exist integers $x$ and $y$, neither of which is divisible by $3$, such that
$x^2+2y^2 = 3^k$.
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
$$\frac1x+\frac1y=\frac1p-\frac1q.$$