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

PEN H Problems, 14
Tags: diophantine equation
Show that the equation $x^2 +y^5 =z^3$ has infinitely many solutions in integers $x, y, z$ for which $xyz \neq 0$.

2013 JBMO Shortlist, 6
Tags: number theory , integer , diophantine equation
Solve in integers the system of equations: $$x^2-y^2=z$$ $$3xy+(x-y)z=z^2$$

2020 Regional Competition For Advanced Students, 4
Tags: diophantine , diophantine equation , number theory
Find all quadruples $(p, q, r, n)$ of prime numbers $p, q, r$ and positive integer numbers $n$, such that $$p^2 = q^2 + r^n$$ (Walther Janous)

1995 Irish Math Olympiad, 2
Tags: number theory , diophantine equation
Determine all integers $ a$ for which the equation $ x^2\plus{}axy\plus{}y^2\equal{}1$ has infinitely many distinct integer solutions $ x,y$.

1998 Croatia National Olympiad, Problem 2
Tags: number theory , diophantine equation
Find all positive integer solutions of the equation $10(m+n)=mn$.

1990 Poland - Second Round, 1
Tags: diophantine , number theory , diophantine equation
Find all pairs of integers $ x $, $ y $ satisfying the equation $$ (xy-1)^2 = (x +1)^2 + (y +1)^2.$$

2002 Singapore Senior Math Olympiad, 3
Tags: number theory , diophantine , radical , diophantine equation
Prove that for natural numbers $p$ and $q$, there exists a natural number $x$ such that $$(\sqrt{p}+\sqrt{p-1})^q=\sqrt{x}+\sqrt{x-1}$$ (As an example, if $p = 3, q = 2$, then $x$ can be taken to be $25$.)

2019 CMI B.Sc. Entrance Exam, 2
Tags: algebra , complex number , diophantine equation
$(a)$ Count the number of roots of $\omega$ of the equation $z^{2019} - 1 = 0 $ over complex numbers that satisfy \begin{align*} \vert \omega + 1 \vert \geq \sqrt{2 + \sqrt{2}} \end{align*} $(b)$ Find all real numbers $x$ that satisfy following equation $:$ \begin{align*} \frac{ 8^x + 27^x }{ 12^x + 18^x } = \frac{7}{6} \end{align*}

2022 New Zealand MO, 1
Tags: number theory , diophantine equation , diophantine
Find all integers $a, b$ such that $$a^2 + b = b^{2022}.$$

PEN H Problems, 24
Tags: geometry , modular arithmetic , diophantine equation , 3d geometry
Prove that if $n$ is a positive integer such that the equation \[x^{3}-3xy^{2}+y^{3}=n.\] has a solution in integers $(x,y),$ then it has at least three such solutions. Show that the equation has no solutions in integers when $n=2891$.

1998 Korea - Final Round, 1
Tags: diophantine equation , relatively prime , number theory , ratio
Find all pairwise relatively prime positive integers $l, m, n$ such that \[(l+m+n)\left( \frac{1}{l}+\frac{1}{m}+\frac{1}{n}\right)\] is an integer.

2013 Czech-Polish-Slovak Junior Match, 1
Tags: number theory , diophantine equation , diophantine , radical
Determine all pairs $(x, y)$ of integers for which satisfy the equality $\sqrt{x-\sqrt{y}}+ \sqrt{x+\sqrt{y}}= \sqrt{xy}$

2009 Postal Coaching, 2
Tags: diophantine , diophantine equation , number theory
Determine, with proof, all the integer solutions of the equation $x^3 + 2y^3 + 4z^3 - 6xyz = 1$.

Mathley 2014-15, 7
Tags: diophantine , prime , number theory , diophantine equation
Find all primes $p,q, r$ such that $\frac{p^{2q}+q^{2p}}{p^3-pq+q^3} = r$. Titu Andreescu, Mathematics Department, College of Texas, USA

2011 Cuba MO, 2
Tags: number theory , diophantine , diophantine equation
Determine all the integer solutions of the equation $3x^4-2024y+1= 0$.

2022 Bulgarian Autumn Math Competition, Problem 9.3
Tags: number theory , diophantine equation , factorial
Find all the pairs of natural numbers $(a, b),$ such that \[a!+1=(a+1)^{(2^b)}\]

2012 France Team Selection Test, 3
Tags: number theory , modular arithmetic , diophantine equation
Let $p$ be a prime number. Find all positive integers $a,b,c\ge 1$ such that: \[a^p+b^p=p^c.\]

2017 Rioplatense Mathematical Olympiad, Level 3, 1
Tags: number theory , diophantine , diophantine equation
Let $a$ be a fixed positive integer. Find the largest integer $b$ such that $(x+a)(x+b)=x+a+b$, for some integer $x$.

2011 Swedish Mathematical Competition, 1
Tags: diophantine , number theory , diophantine equation
Determine all positive integers $k$, $\ell$, $m$ and $n$, such that $$\frac{1}{k!}+\frac{1}{\ell!}+\frac{1}{m!} =\frac{1}{n!} $$

VMEO III 2006, 10.2
Tags: diophantine , number theory , diophantine equation
Find all triples of integers $(x, y, z)$ such that $x^4 + 5y^4 = z^4$.

1987 USAMO, 1
Tags: algebra , diophantine equation , equation
Determine all solutions in non-zero integers $a$ and $b$ of the equation \[(a^2+b)(a+b^2) = (a-b)^3.\]

PEN H Problems, 5
Tags: diophantine equation
Find all pairs $(x, y)$ of rational numbers such that $y^2 =x^3 -3x+2$.

2005 Thailand Mathematical Olympiad, 6
Tags: diophantine , number theory , diophantine equation
Find the number of positive integer solutions to the equation $(x_1+x_2+x_3)^2(y_1+y_2) = 2548$.

1980 Polish MO Finals, 2
Tags: number theory , diophantine , diophantine equation
Prove that for every $n$ there exists a solution of the equation $$a^2 +b^2 +c^2 = 3abc$$ in natural numbers $a,b,c$ greater than $n$.

1995 Austrian-Polish Competition, 7
Tags: number theory , squarefree , perfect square , diophantine , diophantine equation
Consider the equation $3y^4 + 4cy^3 + 2xy + 48 = 0$, where $c$ is an integer parameter. Determine all values of $c$ for which the number of integral solutions $(x,y)$ satisfying the conditions (i) and (ii) is maximal: (i) $|x|$ is a square of an integer; (ii) $y$ is a squarefree number.