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

2011 NZMOC Camp Selection Problems, 4
Tags: diophantine , diophantine equation , number theory
Find all pairs of positive integers $m$ and $n$ such that $$(m + 1)! + (n + 1)! = m^2n.$$

1998 Switzerland Team Selection Test, 2
Tags: diophantine , diophantine equation , number theory
Find all nonnegative integer solutions $(x,y,z)$ of the equation $\frac{1}{x+2}+\frac{1}{y+2}=\frac{1}{2} +\frac{1}{z+2}$

2019 LIMIT Category B, Problem 7
Tags: diophantine equation , number theory
Find the number of ordered pairs of positive integers for which $$\frac1a+\frac1b=\frac4{2019}$$

2016 Finnish National High School Mathematics Comp, 4
Tags: number theory , diophantine equation , diophantine
How many pairs $(a, b)$ of positive integers $a,b$ solutions of the equation $(4a-b)(4b-a )=1770^n$ exist , if $n$ is a positive integer?

2023 Austrian MO Regional Competition, 4
Tags: number theory , gcd , greatest common divisor , diophantine equation
Determine all pairs $(x, y)$ of positive integers such that for $d = gcd(x, y)$ the equation $$xyd = x + y + d^2$$ holds. [i](Walther Janous)[/i]

2003 All-Russian Olympiad Regional Round, 11.1
Tags: number theory , diophantine , diophantine equation
Find all prime $p$, for each of which there are such natural $ x$ and $y$ such that $p^x = y^3 + 1$.

2014 Austria Beginners' Competition, 1
Tags: number theory , diophantine equation , diophantine
Determine all solutions of the diophantine equation $a^2 = b \cdot (b + 7)$ in integers $a\ge 0$ and $b \ge 0$. (W. Janous, Innsbruck)

2015 Cuba MO, 7
Tags: number theory , diophantine , diophantine equation
If $p$ is a prime number and $x, y$ are positive integers, find in terms of $p$, all pairs $(x, y)$ that satisfy the equation: $$p(x -2) = x(y -1).$$ If $x+y = 21$, find all triples $(x, y, p)$ that satisfy this equation.

PEN H Problems, 80
Tags: diophantine equation
Prove that if $a, b, c, d$ are integers such that $d=( a+\sqrt[3]{2}b+\sqrt[3]{4}c)^{2}$ then $d$ is a perfect square.

1999 Estonia National Olympiad, 1
Tags: diophantine , diophantine equation , number theory
Find all pairs of integers $(m, n)$ such that $(m - n)^2 =\frac{4mn}{m + n - 1}$

Mathley 2014-15, 2
Tags: diophantine equation , number theory , diophantine
Let $n$ be a positive integer and $p$ a prime number $p > n + 1$. Prove that the following equation does not have integer solution $$1 + \frac{x}{n + 1} + \frac{x^2}{2n + 1} + ...+ \frac{x^p}{pn + 1} = 0$$ Luu Ba Thang, Department of Mathematics, College of Education

PEN H Problems, 85
Tags: diophantine equation , inequalities
Find all integer solutions to $2(x^5 +y^5 +1)=5xy(x^2 +y^2 +1)$.

2006 IMO Shortlist, 1
Tags: number theory , diophantine equation
Determine all pairs $(x, y)$ of integers such that \[1+2^{x}+2^{2x+1}= y^{2}.\]

1997 Brazil Team Selection Test, Problem 3
Tags: diophantine equation , number theory
Let $b$ be a positive integer such that $\gcd(b,6)=1$. Show that there are positive integers $x$ and $y$ such that $\frac1x+\frac1y=\frac3b$ if and only if $b$ is divisible by some prime number of form $6k-1$.

2022 Lusophon Mathematical Olympiad, 4
Tags: number theory , diophantine equation
How many integer solutions exist that satisfy this equation? $$x+4y-343\sqrt{x}-686\sqrt{y}+4\sqrt{xy}+2022=0$$.

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

2012 AMC 10, 22
Tags: modular arithmetic , number theory , diophantine equation , system of equations , algebra
The sum of the first $m$ positive odd integers is $212$ more than the sum of the first $n$ positive even integers. What is the sum of all possible values of $n$? $ \textbf{(A)}\ 255 \qquad\textbf{(B)}\ 256 \qquad\textbf{(C)}\ 257 \qquad\textbf{(D)}\ 258 \qquad\textbf{(E)}\ 259 $

PEN H Problems, 1
Tags: diophantine equation
One of Euler's conjectures was disproved in the $1980$s by three American Mathematicians when they showed that there is a positive integer $n$ such that \[n^{5}= 133^{5}+110^{5}+84^{5}+27^{5}.\] Find the value of $n$.

2002 Junior Balkan Team Selection Tests - Romania, 2
Tags: number theory , diophantine , diophantine equation
Find all positive integers $a, b,c,d$ such that $a + b + c + d - 3 = ab + cd$.

2014 Peru IMO TST, 9
Tags: diophantine equation , number theory
Prove that for every positive integer $n$ there exist integers $a$ and $b,$ both greater than $1,$ such that $a ^ 2 + 1 = 2b ^ 2$ and $a - b$ is a multiple of $n.$

2022 ELMO Revenge, 2
Tags: number theory , diophantine equation
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]

2011 Dutch Mathematical Olympiad, 1
Tags: number theory , factorial , diophantine , diophantine equation
Determine all triples of positive integers $(a, b, n)$ that satisfy the following equation: $a! + b! = 2^n$

PEN H Problems, 41
Tags: diophantine equation , geometry , 3d geometry , number theory , relatively prime
Suppose that $A=1,2,$ or $3$. Let $a$ and $b$ be relatively prime integers such that $a^{2}+Ab^2 =s^3$ for some integer $s$. Then, there are integers $u$ and $v$ such that $s=u^2 +Av^2$, $a =u^3 - 3Avu^2$, and $b=3u^{2}v -Av^3$.

2016 District Olympiad, 1
Tags: equation , diophantine equation , algebra
Solve in $ \mathbb{N}^2: $ $$ x+y=\sqrt x+\sqrt y+\sqrt{xy} . $$

2013 Dutch BxMO/EGMO TST, 3
Tags: number theory , prime , diophantine equation
Find all triples $(x,n,p)$ of positive integers $x$ and $n$ and primes $p$ for which the following holds $x^3 + 3x + 14 = 2 p^n$