Found problems: 436
2022 Dutch BxMO TST, 3
Find all pairs $(p, q)$ of prime numbers such that $$p(p^2 -p - 1) = q(2q + 3).$$
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.
2014 Dutch Mathematical Olympiad, 1
Determine all triples $(a,b,c)$, where $a, b$, and $c$ are positive integers that satisfy
$a \le b \le c$ and $abc = 2(a + b + c)$.
1971 Dutch Mathematical Olympiad, 3
Prove that $(0,1)$, $(0, -1)$,$( -1,1)$ and $(-1,-1)$ are the only integer solutions of $$x^2 + x +1 = y^2.$$
2017 District Olympiad, 2
Let $ E(x,y)=\frac{x}{y} +\frac{x+1}{y+1} +\frac{x+2}{y+2} . $
[b]a)[/b] Solve in $ \mathbb{N}^2 $ the equation $ E(x,y)=3. $
[b]b)[/b] Show that there are infinitely many natural numbers $ n $ such that the equation $ E(x,y)=n $ has at least one solution in $ \mathbb{N}^2. $
2021 Bosnia and Herzegovina Junior BMO TST, 2
Let $p, q, r$ be prime numbers and $t, n$ be natural numbers such that $p^2 +qt =(p + t)^n$ and $p^2 + qr = t^4$ .
a) Show that $n < 3$.
b) Determine all the numbers $p, q, r, t, n$ that satisfy the given conditions.
2018 India PRMO, 6
Integers $a, b, c$ satisfy $a+b-c=1$ and $a^2+b^2-c^2=-1$. What is the sum of all possible values of $a^2+b^2+c^2$ ?
2005 iTest, 34
If $x$ is the number of solutions to the equation $a^2 + b^2 + c^2 = d^2$ of the form $(a,b,c,d)$ such that $\{a,b,c\}$ are three consecutive square numbers and $d$ is also a square number, find $x$.
1999 Estonia National Olympiad, 1
Find all pairs of integers ($a, b$) such that $a^2 + b = b^{1999}$ .
2011 Bundeswettbewerb Mathematik, 4
Let $a$ and $b$ be positive integers. As is known, the division of of $a \cdot b$ with $a + b$ determines integers $q$ and $r$ uniquely such that $a \cdot b = q (a + b) + r$ and $0 \le r <a + b$. Find all pairs $(a, b)$ for which $q^2 + r = 2011$.
2010 Regional Olympiad of Mexico Center Zone, 5
Find all integer solutions $(p, q, r)$ of the equation $r + p ^ 4 = q ^ 4$ with the following conditions:
$\bullet$ $r$ is a positive integer with exactly $8$ positive divisors.
$\bullet$ $p$ and $q$ are prime numbers.
1990 Austrian-Polish Competition, 2
Find all solutions in positive integers to $a^A = b^B = c^C = 1990^{1990}abc$, where $A = b^c, B = c^a, C = a^b$.
2016 Costa Rica - Final Round, N1
Let $p> 5$ be a prime such that none of its digits is divisible by $3$ or $7$. Prove that the equation $x^4 + p = 3y^4$ does not have integer solutions.
2009 Philippine MO, 2
[b](a)[/b] Find all pairs $(n,x)$ of positive integers that satisfy the equation $2^n + 1 = x^2$.
[b](b)[/b] Find all pairs $(n,x)$ of positive integers that satisfy the equation $2^n = x^2 + 1$.
2016 Costa Rica - Final Round, N3
Find all natural values of $n$ and $m$, such that $(n -1)2^{n - 1} + 5 = m^2 + 4m$.
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 (*)?
VI Soros Olympiad 1999 - 2000 (Russia), 11.4
For prime numbers $p$ and $q$, natural numbers $n$, $k$, $r$, the equality $p^{2k}+q^{2n}=r^2$ holds. Prove that the number $r$ is prime.
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.
1998 Singapore MO Open, 3
Do there exist integers $x$ and $y$ such that $19^{19} = x^3 +y^4$ ? Justify your answer.
2005 Estonia Team Selection Test, 3
Find all pairs $(x, y)$ of positive integers satisfying the equation $(x + y)^x = x^y$.
1967 Swedish Mathematical Competition, 3
Show that there are only finitely many triples $(a, b, c)$ of positive integers such that $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{1000}$.
2022 Austrian MO Beginners' Competition, 4
Determine all prime numbers $p, q$ and $r$ with $p + q^2 = r^4$.
[i](Karl Czakler)[/i]
2015 NZMOC Camp Selection Problems, 4
For which positive integers $m$ does the equation: $$(ab)^{2015} = (a^2 + b^2)^m$$ have positive integer solutions?
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
1949 Moscow Mathematical Olympiad, 158
a) Prove that $x^2 + y^2 + z^2 = 2xyz$ for integer $x, y, z$ only if $x = y = z = 0$.
b) Find integers $x, y, z, u$ such that $x^2 + y^2 + z^2 + u^2 = 2xyzu$.