Found problems: 916
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.
PEN H Problems, 81
Find a pair of relatively prime four digit natural numbers $A$ and $B$ such that for all natural numbers $m$ and $n$, $\vert A^m -B^n \vert \ge 400$.
2018 Korea Junior Math Olympiad, 4
For a positive integer $n$, denote $p(n)$ to be the number of nonnegative integer tuples $(x,y,z,w)$ such that $x+2y+2z+3w=n$. Also, denote $q(n)$ to be the number of nonnegative integer tuples $(a,b,c,d)$ such that
(i) $a+b+c+d=n$
(ii) $a \ge b \ge d$
(iii) $a \ge c \ge d$
Prove that for all $n$, $p(n) = q(n)$.
PEN H Problems, 12
Find all $(x,y,z) \in {\mathbb{N}}^3$ such that $x^{4}-y^{4}=z^{2}$.
2025 Bangladesh Mathematical Olympiad, P4
Find all prime numbers $p, q$ such that$$p(p+1)(p^2+1) = q^2(q^2+q+1) + 2025.$$
[i]Proposed by Md. Fuad Al Alam[/i]
2014 Contests, 2
$p$ is a prime. Find the all $(m,n,p)$ positive integer triples satisfy $m^3+7p^2=2^n$.
2002 Romania Team Selection Test, 2
The sequence $ (a_n)$ is defined by: $ a_0\equal{}a_1\equal{}1$ and $ a_{n\plus{}1}\equal{}14a_n\minus{}a_{n\minus{}1}$ for all $ n\ge 1$.
Prove that $ 2a_n\minus{}1$ is a perfect square for any $ n\ge 0$.
2019 Azerbaijan Senior NMO, 3
Find all $x;y\in\mathbb{Z}$ satisfying the following condition: $$x^3=y^4+9x^2$$
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.
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 (*)?
PEN H Problems, 62
Solve the equation $7^x -3^y =4$ in positive integers.
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 $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{1000}$.
1991 Irish Math Olympiad, 1
Problem. The sum of two consecutive squares can be a square; for instance $3^2 + 4^2 = 5^2$.
(a) Prove that the sum of $m$ consecutive squares cannot be a square for $m \in \{3, 4, 5, 6\}$.
(b) Find an example of eleven consecutive squares whose sum is a square.
Can anyone help me with this?
Thanks.
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?
PEN H Problems, 26
Solve in integers the following equation \[n^{2002}=m(m+n)(m+2n)\cdots(m+2001n).\]
2022 USAMO, 4
Find all pairs of primes $(p, q)$ for which $p-q$ and $pq-q$ are both perfect squares.