This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 916

2016 NZMOC Camp Selection Problems, 7
Tags: diophantine , diophantine equation , number theory
Find all positive integers $n$ for which the equation $$(x^2 + y^2)^n = (xy)^{2016}$$ has positive integer solutions.

2018 Hanoi Open Mathematics Competitions, 11
Tags: diophantine equation , number theory
Find all pairs of nonnegative integers $(x, y)$ for which $(xy + 2)^2 = x^2 + y^2 $.

PEN H Problems, 49
Tags: search , group theory , abstract algebra , diophantine equation
Show that the only solutions of the equation $x^{3}-3xy^2 -y^3 =1$ are given by $(x,y)=(1,0),(0,-1),(-1,1),(1,-3),(-3,2),(2,1)$.

PEN H Problems, 40
Tags: diophantine equation
Determine all pairs of rational numbers $(x, y)$ such that \[x^{3}+y^{3}= x^{2}+y^{2}.\]

2005 iTest, 16
Tags: diophantine , number theory , diophantine equation
How many distinct integral solutions of the form $(x, y)$ exist to the equation$ 21x + 22y = 43$ such that $1 < x < 11$ and $y < 22$?

II Soros Olympiad 1995 - 96 (Russia), 9.2
Tags: diophantine equation , diophantine , number theory
Find the integers $x, y, z$ for which $$\dfrac{1}{x+\dfrac{1}{y+\dfrac{1}{z}}}=\dfrac{7}{17}$$

PEN H Problems, 52
Tags: diophantine equation
Do there exist two right-angled triangles with integer length sides that have the lengths of exactly two sides in common?

2015 Turkey Junior National Olympiad, 3
Tags: number theory , diophantine equation , prime number
Find all pairs $(p,n)$ so that $p$ is a prime number, $n$ is a positive integer and \[p^3-2p^2+p+1=3^n \] holds.

2019 Romania National Olympiad, 4
Tags: number theory , diophantine equation , diophantine
Find the natural numbers $x, y, z$ that verify the equation: $$2^x + 3 \cdot 11^y =7^z$$

1980 IMO Longlists, 12
Tags: number theory , diophantine equation , algebra , equation , polynomial
Find all pairs of solutions $(x,y)$: \[ x^3 + x^2y + xy^2 + y^3 = 8(x^2 + xy + y^2 + 1). \]

2000 Estonia National Olympiad, 3
Tags: number theory , diophantine , diophantine equation
Are there any (not necessarily positive) integers $m$ and $n$ such that a) $\frac{1}{m}-\frac{1}{n}=\frac{1}{m-n}$ ? b) $\frac{1}{m}-\frac{1}{n}=\frac{1}{n-m}$

PEN H Problems, 39
Tags: quadratic , diophantine equation
Let $A, B, C, D, E$ be integers, $B \neq 0$ and $F=AD^{2}-BCD+B^{2}E \neq 0$. Prove that the number $N$ of pairs of integers $(x, y)$ such that \[Ax^{2}+Bxy+Cx+Dy+E=0,\] satisfies $N \le 2 d( \vert F \vert )$, where $d(n)$ denotes the number of positive divisors of positive integer $n$.

2018 Regional Olympiad of Mexico Northeast, 3
Tags: diophantine equation , number theory
Find the smallest natural number $n$ for which there exists a natural number $x$ such that $$(x+1)^3 + (x + 2)^3 + (x + 3)^3 + (x + 4)^3 = (x + n)^3.$$

1989 IMO Longlists, 8
Tags: algebra , polynomial , root , equation , diophantine equation
Find the roots $ r_i \in \mathbb{R}$ of the polynomial \[ p(x) \equal{} x^n \plus{} n \cdot x^{n\minus{}1} \plus{} a_2 \cdot x^{n\minus{}2} \plus{} \ldots \plus{} a_n\] satisfying \[ \sum^{16}_{k\equal{}1} r^{16}_k \equal{} n.\]

2011 Greece JBMO TST, 3
Tags: number theory , diophantine equation , integer
Find integer solutions of the equation $8x^3 - 4 = y(6x - y^2)$

1973 Chisinau City MO, 70
Tags: diophantine , diophantine equation , number theory
The natural numbers $p, q$ satisfy the relation $p^p + q^q = p^q + q^p$. Prove that $p = q$.

2003 Turkey MO (2nd round), 1
Tags: greatest common divisor , modular arithmetic , diophantine equation , number theory
Suppose that $2^{2n+1}+ 2^{n}+1=x^{k}$, where $k\geq2$ and $n$ are positive integers. Find all possible values of $n$.

2015 China Team Selection Test, 5
Tags: number theory , diophantine equation
FIx positive integer $n$. Prove: For any positive integers $a,b,c$ not exceeding $3n^2+4n$, there exist integers $x,y,z$ with absolute value not exceeding $2n$ and not all $0$, such that $ax+by+cz=0$

1998 Bundeswettbewerb Mathematik, 1
Tags: diophantine equation , diophantine , number theory
Find all integer solutions $(x,y,z)$ of the equation $xy+yz+zx-xyz = 2$.

2020 Chile National Olympiad, 4
Tags: number theory , diophantine , system of equations , diophantine equation
Determine all three integers $(x, y, z)$ that are solutions of the system $$x + y -z = 6$$ $$x^3 + y^3 -z^3 = 414$$

2010 District Olympiad, 4
Tags: diophantine equation , number theory , diophantine
Find all non negative integers $(a, b)$ such that $$a + 2b - b^2 =\sqrt{2a + a^2 + |2a + 1 - 2b|}.$$

2018 Ecuador NMO (OMEC), 1
Tags: number theory , diophantine equation , diophantine
Let $a, b$ be integers. Show that the equation $a^2 + b^2 = 26a$ has at least $12$ solutions.

2014 Belarus Team Selection Test, 4
Tags: number theory , diophantine , diophantine equation
Find all integers $a$ and $b$ satisfying the equality $3^a - 5^b = 2$. (I. Gorodnin)

2016 Abels Math Contest (Norwegian MO) Final, 2a
Tags: diophantine , number theory , system of equations , diophantine equation
Find all positive integers $a, b, c, d$ with $a \le b$ and $c \le d$ such that $\begin{cases} a + b = cd \\ c + d = ab \end{cases}$ .

1990 IMO Longlists, 26
Tags: modular arithmetic , search , diophantine equation , number theory
Prove that there exist infinitely many positive integers $n$ such that the number $\frac{1^2+2^2+\cdots+n^2}{n}$ is a perfect square. Obviously, $1$ is the least integer having this property. Find the next two least integers having this property.