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.

AND:
OR:
NO:

Found problems: 916

2018 Regional Olympiad of Mexico Center Zone, 5
Tags: diophantine equation , diophantine , number theory
Find all solutions of the equation $$p ^ 2 + q ^ 2 + 49r ^ 2 = 9k ^ 2-101$$ with $ p$, $q$ and $r$ positive prime numbers and $k$ a positive integer.

2013 India Regional Mathematical Olympiad, 6
Tags: quadratic , modular arithmetic , diophantine equation , number theory
Suppose that $m$ and $n$ are integers, such that both the quadratic equations $x^2+mx-n=0$ and $x^2-mx+n=0$ have integer roots. Prove that $n$ is divisible by $6$.

1995 Tuymaada Olympiad, 3
Tags: diophantine equation , algebra , number theory
Prove that the equation $(\sqrt5 +1)^{2x}+ (\sqrt5 -1)^{2x}=2^x(y^2+2)$ has an infinite number of solutions in natural numbers.

PEN H Problems, 31
Tags: diophantine equation , ring theory
Determine all integer solutions of the system \[2uv-xy=16,\] \[xv-yu=12.\]

PEN H Problems, 9
Tags: diophantine equation , algebra , polynomial , vieta , absolute value
Determine all integers $a$ for which the equation \[x^{2}+axy+y^{2}=1\] has infinitely many distinct integer solutions $x, \;y$.

2023 Bangladesh Mathematical Olympiad, P1
Tags: factorial , number theory , diophantine equation
Find all possible non-negative integer solution ($x,$ $y$) of the following equation- $$x!+2^y=z!$$ Note: $x!=x\cdot(x-1)!$ and $0!=1$. For example, $5!=5\times4\times3\times2\times1=120$.

2018 Belarusian National Olympiad, 10.5
Tags: diophantine equation , number theory
Find all positive integers $n$ such that equation $$3a^2-b^2=2018^n$$ has a solution in integers $a$ and $b$.

1976 Swedish Mathematical Competition, 6
Tags: diophantine , diophantine equation , number theory
Show that there are only finitely many integral solutions to \[ 3^m - 1 = 2^n \] and find them.

2010 Czech And Slovak Olympiad III A, 1
Tags: number theory , diophantine equation
Determine all pairs of integers $a, b$ for which they apply $4^a + 4a^2 + 4 = b^2$ .

PEN H Problems, 58
Tags: diophantine equation , modular arithmetic , algorithm
Solve in positive integers the equation $10^{a}+2^{b}-3^{c}=1997$.

2019 Hanoi Open Mathematics Competitions, 11
Tags: number theory , diophantine , diophantine equation
Find all integers $x$ and $y$ satisfying the following equation $x^2 - 2xy + 5y^2 + 2x - 6y - 3 = 0$.

PEN H Problems, 48
Tags: diophantine equation , modular arithmetic , search
Solve the equation $x^2 +7=2^n$ in integers.

1982 IMO, 1
Tags: number theory , equation , diophantine equation , divisibility
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 for $n=2891$.

2022 IMO, 5
Tags: number theory , divisibility , lte lemma , diophantine equation
Find all triples $(a,b,p)$ of positive integers with $p$ prime and \[ a^p=b!+p. \]

2023 Israel National Olympiad, P2
Tags: number theory , diophantine equation
The non-negative integers $x,y$ satisfy $\sqrt{x}+\sqrt{x+60}=\sqrt{y}$. Find the largest possible value for $x$.

2001 JBMO ShortLists, 1
Tags: geometry , 3d geometry , diophantine equation , number theory
Find the positive integers $n$ that are not divisible by $3$ if the number $2^{n^2-10}+2133$ is a perfect cube. [hide="Note"] [color=#BF0000]The wording of this problem is perhaps not the best English. As far as I am aware, just solve the diophantine equation $x^3=2^{n^2-10}+2133$ where $x,n \in \mathbb{N}$ and $3\nmid n$.[/color][/hide]

2016 Costa Rica - Final Round, N3
Tags: number theory , diophantine equation , diophantine
Find all nonnegative integers $a$ and $b$ that satisfy the equation $$3 \cdot 2^a + 1 = b^2.$$

2007 QEDMO 4th, 8
Tags: number theory , diophantine equation , diophantine
Show that there are no integers $x$ and $y$ satisfying $x^2 + 5 = y^3$. Daniel Harrer

2008 Bulgarian Autumn Math Competition, Problem 10.3
Tags: diophantine equation , number theory
Find all natural numbers $x,y,z$, such that $7^{x}+13^{y}=2^{z}$.

2009 Postal Coaching, 2
Tags: diophantine , diophantine equation , number theory
Solve for prime numbers $p, q, r$ : $$\frac{p}{q} - \frac{4}{r + 1}= 1$$

PEN H Problems, 46
Tags: diophantine equation , calculus , integration
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.

2017 Swedish Mathematical Competition, 2
Tags: number theory , diophantine equation , diophantine
Let $p$ be a prime number. Find all pairs of coprime positive integers $(m,n)$ such that $$ \frac{p+m}{p+n}=\frac{m}{n}+\frac{1}{p^2}.$$

2011 Chile National Olympiad, 1
Tags: number theory , diophantine , diophantine equation
Find all the solutions $(a, b, c)$ in the natural numbers, verifying $1\le a \le b \le c$, of the equation$$\frac34=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}.$$

1966 Bulgaria National Olympiad, Problem 1
Tags: number theory , diophantine equation
Prove that the equation $$3x(x-3y)=y^2+z^2$$doesn't have any integer solutions except $x=0,y=0,z=0$.

2011 Junior Balkan Team Selection Tests - Romania, 4
Tags: system of equations , diophantine equation , diophantine , number theory
Show that there is an infinite number of positive integers $t$ such that none of the equations $$ \begin{cases} x^2 + y^6 = t \\ x^2 + y^6 = t + 1 \\ x^2 - y^6 = t \\ x^2 - y^6 = t + 1 \end{cases}$$ has solutions $(x, y) \in Z \times Z$.