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

2024 Greece Junior Math Olympiad, 4
Tags: number theory , diophantine equation , diophantine
Prove that there are infinite triples of positive integers $(x,y,z)$ such that $$x^2+y^2+z^2+xy+yz+zx=6xyz.$$

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

2016 Croatia Team Selection Test, Problem 4
Tags: number theory , prime , diophantine equation , prime number
Find all pairs $(p,q)$ of prime numbers such that $$ p(p^2 - p - 1) = q(2q + 3) .$$

2008 Hanoi Open Mathematics Competitions, 2
Tags: number theory , diophantine equation , diophantine
Find all pairs $(m, n)$ of positive integers such that $m^2 + 2n^2 = 3(m + 2n)$

1980 IMO Shortlist, 3
Tags: number theory , equation , diophantine equation , algebra
Prove that the equation \[ x^n + 1 = y^{n+1}, \] where $n$ is a positive integer not smaller then 2, has no positive integer solutions in $x$ and $y$ for which $x$ and $n+1$ are relatively prime.

2018 Spain Mathematical Olympiad, 5
Tags: number theory , diophantine equation
Let $a, b$ be coprime positive integers. A positive integer $n$ is said to be [i]weak[/i] if there do not exist any nonnegative integers $x, y$ such that $ax+by=n$. Prove that if $n$ is a [i]weak[/i] integer and $n < \frac{ab}{6}$, then there exists an integer $k \geq 2$ such that $kn$ is [i]weak[/i].

2012 Dutch IMO TST, 3
Tags: number theory , diophantine equation , diophantine
Determine all pairs $(x, y)$ of positive integers satisfying $x + y + 1 | 2xy$ and $ x + y - 1 | x^2 + y^2 - 1$.

2018 NZMOC Camp Selection Problems, 2
Tags: diophantine , diophantine equation , number theory
Find all pairs of integers $(a, b)$ such that $$a^2 + ab - b = 2018.$$

VMEO III 2006 Shortlist, N5
Tags: number theory , diophantine equation , diophantine
Find all triples of integers $(x, y, z)$ such that $x^4 + 5y^4 = z^4$.

2012 AMC 10, 22
Tags: system of equations , algebra , number theory , modular arithmetic , diophantine equation
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 $

2020-IMOC, N2
Tags: number theory , diophantine equation
Find all positive integers $N$ such that the following holds: There exist pairwise coprime positive integers $a,b,c$ with $$\frac1a+\frac1b+\frac1c=\frac N{a+b+c}.$$

PEN H Problems, 19
Tags: diophantine equation
Find all $(x, y, z, n) \in {\mathbb{N}}^4$ such that $ x^3 +y^3 +z^3 =nx^2 y^2 z^2$.

2022 Austrian MO National Competition, 4
Tags: prime number , diophantine equation , diophantine , number theory
Find all triples $(p, q, r)$ of prime numbers for which $4q - 1$ is a prime number and $$\frac{p + q}{p + r} = r - p$$ holds. [i](Walther Janous)[/i]

2021 Federal Competition For Advanced Students, P2, 3
Tags: number theory , diophantine equation , diophantine
Find all triples $(a, b, c)$ of natural numbers $a, b$ and $c$, for which $a^{b + 20} (c-1) = c^{b + 21} - 1$ is satisfied. (Walther Janous)

2024 Ecuador NMO (OMEC), 5
Tags: number theory , diophantine equation
Find all triples of non-negative integer numbers $(E, C, U)$ such that $EC \ge 1$ and: $$2^{3^E}+3^{2^C}=593 \cdot 5^U$$

2007 Indonesia TST, 3
Tags: number theory , floor function , diophantine equation , inequalities
For each real number $ x$< let $ \lfloor x \rfloor$ be the integer satisfying $ \lfloor x \rfloor \le x < \lfloor x \rfloor \plus{}1$ and let $ \{x\}\equal{}x\minus{}\lfloor x \rfloor$. Let $ c$ be a real number such that \[ \{n\sqrt{3}\}>\dfrac{c}{n\sqrt{3}}\] for all positive integers $ n$. Prove that $ c \le 1$.

PEN H Problems, 79
Tags: modular arithmetic , diophantine equation
Find all positive integers $m$ and $n$ for which \[1!+2!+3!+\cdots+n!=m^{2}\]

1992 Bulgaria National Olympiad, Problem 4
Tags: diophantine equation , number theory
Let $p$ be a prime number in the form $p=4k+3$. Prove that if the numbers $x_0,y_0,z_0,t_0$ are solutions of the equation $x^{2p}+y^{2p}+z^{2p}=t^{2p}$, then at least one of them is divisible by $p$. [i](Plamen Koshlukov)[/i]

2004 Argentina National Olympiad, 2
Tags: diophantine equation , system of equations , number theory
Determine all positive integers $a,b,c,d$ such that$$\begin{cases} a<b \\ a^2c =b^2d \\ ab+cd =2^{99}+2^{101} \end{cases}$$

1984 IMO Shortlist, 2
Tags: polynomial , number theory , diophantine equation , equation , quadratic
Prove: (a) There are infinitely many triples of positive integers $m, n, p$ such that $4mn - m- n = p^2 - 1.$ (b) There are no positive integers $m, n, p$ such that $4mn - m- n = p^2.$

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

PEN H Problems, 8
Tags: integration , calculus , diophantine equation
Show that the equation \[x^{3}+y^{3}+z^{3}+t^{3}=1999\] has infinitely many integral solutions.

2010 Junior Balkan Team Selection Tests - Romania, 1
Tags: number theory , diophantine equation , diophantine
Determine the prime numbers $p, q, r$ with the property that: $p(p-7) + q (q-7) = r (r-7)$.

2025 6th Memorial "Aleksandar Blazhevski-Cane", P1
Tags: number theory , prime number , diophantine equation
Determine all triples of prime numbers $(p, q, r)$ that satisfy \[p2^q + r^2 = 2025.\] Proposed by [i]Ilija Jovcevski[/i]