Found problems: 916
2023 German National Olympiad, 1
Determine all pairs $(m,n)$ of integers with $n \ge m$ satisfying the equation
\[n^3+m^3-nm(n+m)=2023.\]
2011 Mongolia Team Selection Test, 1
Let $A=\{a^2+13b^2 \mid a,b \in\mathbb{Z}, b\neq0\}$. Prove that there
a) exist
b) exist infinitely many
$x,y$ integer pairs such that $x^{13}+y^{13} \in A$ and $x+y \notin A$.
(proposed by B. Bayarjargal)
2024 Polish MO Finals, 3
Determine all pairs $(p,q)$ of prime numbers with the following property: There are positive integers $a,b,c$ satisfying
\[\frac{p}{a}+\frac{p}{b}+\frac{p}{c}=1 \quad \text{and} \quad \frac{a}{p}+\frac{b}{p}+\frac{c}{p}=q+1.\]
PEN H Problems, 21
Prove that the equation \[6(6a^{2}+3b^{2}+c^{2}) = 5n^{2}\] has no solutions in integers except $a=b=c=n=0$.
PEN H Problems, 46
Let $a, b, c, d, e, f$ be integers such that $b^{2}-4ac>0$ is not a perfect square and $4acf+bde-ae^{2}-cd^{2}-fb^{2}\neq 0$. Let \[f(x, y)=ax^{2}+bxy+cy^{2}+dx+ey+f\] Suppose that $f(x, y)=0$ has an integral solution. Show that $f(x, y)=0$ has infinitely many integral solutions.
PEN H Problems, 50
Show that the equation $y^{2}=x^{3}+2a^{3}-3b^2$ has no solution in integers if $ab \neq 0$, $a \not\equiv 1 \; \pmod{3}$, $3$ does not divide $b$, $a$ is odd if $b$ is even, and $p=t^2 +27u^2$ has a solution in integers $t,u$ if $p \vert a$ and $p \equiv 1 \; \pmod{3}$.
2019 District Olympiad, 1
Determine the integers $a, b, c$ for which
$$\frac{a+1}{3}=\frac{b+2}{4}=\frac{5}{c+3}$$
1998 Estonia National Olympiad, 4
Find all integers $n > 2$ for which $(2n)! = (n-2)!n!(n+2)!$ .
2005 Taiwan TST Round 1, 1
Prove that there exists infinitely many positive integers $n$ such that $n, n+1$, and $n+2$ can be written as the sum of two perfect squares.
2021 New Zealand MO, 4
Find all triples $(x, p, n)$ of non-negative integers such that $p$ is prime and $2x(x + 5) = p^n + 3(x - 1)$.
2008 AIME Problems, 4
There exist unique positive integers $ x$ and $ y$ that satisfy the equation $ x^2 \plus{} 84x \plus{} 2008 \equal{} y^2$. Find $ x \plus{} y$.
1987 Dutch Mathematical Olympiad, 1
Solve into $N$: $$a^2 = 2^b +c^4$$
2005 Austrian-Polish Competition, 9
Consider the equation $x^3 + y^3 + z^3 = 2$.
a) Prove that it has infinitely many integer solutions $x,y,z$.
b) Determine all integer solutions $x, y, z$ with $|x|, |y|, |z| \leq 28$.
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
PEN H Problems, 51
Prove that the product of five consecutive positive integers is never a perfect square.
2014 Junior Balkan Team Selection Tests - Romania, 2
Determine all pairs $(a, b)$ of integers which satisfy the equality $\frac{a + 2}{b + 1} +\frac{a + 1}{b + 2} = 1 +\frac{6}{a + b + 1}$
2018 Regional Olympiad of Mexico Northeast, 3
Find the smallest natural number $n$ for which there exists a natural number $x$ such that
$$(x+1)^3 + (x + 2)^3 + (x + 3)^3 + (x + 4)^3 = (x + n)^3.$$
2021 Nigerian MO Round 3, Problem 3
Find all pairs of natural numbers $(p, n)$ with $p$ prime such that $p^6+p^5+n^3+n=n^5+n^2$.
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.
2015 USAMO, 5
Let $a$, $b$, $c$, $d$, $e$ be distinct positive integers such that $a^4+b^4=c^4+d^4=e^5$. Show that $ac+bd$ is a composite number.
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
2022 ELMO Revenge, 2
Find all ordered pairs of integers $x,y$ such that $$xy(x^2y^2 - 12xy- 12x- 12y+2) = (2x + 2y)^2.$$
[i]Proposed by Henry Jiang[/i]
2001 Croatia National Olympiad, Problem 1
Find all integers $x$ for which $2x^2-x-36$ is the square of a prime number.
PEN H Problems, 65
Determine all pairs $(x, y)$ of integers such that \[(19a+b)^{18}+(a+b)^{18}+(19b+a)^{18}\] is a nonzero perfect square.
2011 Croatia Team Selection Test, 4
Find all pairs of integers $x,y$ for which
\[x^3+x^2+x=y^2+y.\]