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, 83
Tags: diophantine equation
Find all pairs $(a, b)$ of positive integers such that \[(\sqrt[3]{a}+\sqrt[3]{b}-1 )^{2}= 49+20 \sqrt[3]{6}.\]

2010 Malaysia National Olympiad, 6
Tags: diophantine equation , algebra
Find the smallest integer $k\ge3$ with the property that it is possible to choose two of the number $1,2,...,k$ in such a way that their product is equal to the sum of the remaining $k-2$ numbers.

2022 Federal Competition For Advanced Students, P1, 4
Tags: prime number , number theory , diophantine equation , diophantine
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]

1989 IMO Longlists, 7
Tags: algebra , equation , polynomial , diophantine equation , rational roots
Prove that $ \forall n > 1, n \in \mathbb{N}$ the equation \[ \sum^n_{k\equal{}1} \frac{x^k}{k!} \plus{} 1 \equal{} 0\] has no rational roots.

2007 Moldova Team Selection Test, 2
Tags: algebra , polynomial , search , diophantine equation , number theory
Find all polynomials $f\in \mathbb{Z}[X]$ such that if $p$ is prime then $f(p)$ is also prime.

1997 Singapore MO Open, 3
Tags: factor , number theory , diophantine equation , diophantine , divisor
Find all the natural numbers $N$ which satisfy the following properties: (i) $N$ has exactly $6$ distinct factors $1, d_1, d_2, d_3, d_4, N$ and (ii) $1 + N = 5(d_1 + d_2+d_3 + d_4)$. Justify your answers.

2013 Swedish Mathematical Competition, 3
Tags: diophantine equation , diophantine , number theory
Determine all primes $p$ and all non-negative integers $m$ and $n$, such that $$1 + p^n = m^3. $$

2013 Hanoi Open Mathematics Competitions, 5
Tags: number theory , diophantine equation , diophantine
The number of integer solutions $x$ of the equation below $(12x -1)(6x - 1)(4x -1)(3x - 1) = 330$ is (A): $0$, (B): $1$, (C): $2$, (D): $3$, (E): None of the above.

1993 Austrian-Polish Competition, 1
Tags: diophantine equation , diophantine , number theory
Solve in positive integers $x,y$ the equation $2^x - 3^y = 7$.

2016 Belarus Team Selection Test, 3
Tags: number theory , diophantine equation
Solve the equation $p^3-q^3=pq^3-1$ in primes $p,q$.

1991 Swedish Mathematical Competition, 1
Tags: diophantine equation , diophantine , number theory
Find all positive integers $m, n$ such that $\frac{1}{m} + \frac{1}{n} - \frac{1}{mn} =\frac{2}{5}$.

PEN H Problems, 6
Tags: diophantine equation , number theory
Show that there are infinitely many pairs $(x, y)$ of rational numbers such that $x^3 +y^3 =9$.

2016 Indonesia MO, 2
Tags: diophantine equation , number theory
Determine all triples of natural numbers $(a,b, c)$ with $b> 1$ such that $2^c + 2^{2016} = a^b$.

2007 Abels Math Contest (Norwegian MO) Final, 3
Tags: diophantine equation , diophantine , number theory
(a) Let $x$ and $y$ be two positive integers such that $\sqrt{x} +\sqrt{y}$ is an integer. Show that $\sqrt{x}$ and $\sqrt{y}$ are both integers. (b) Find all positive integers $x$ and $y$ such that $\sqrt{x} +\sqrt{y}=\sqrt{2007}$.

2000 Switzerland Team Selection Test, 7
Tags: diophantine equation , diophantine , number theory
Show that the equation $14x^2 +15y^2 = 7^{2000}$ has no integer solutions.

1977 Dutch Mathematical Olympiad, 1
Tags: number theory , diophantine , diophantine equation
Solve, for integers $x$ and $y$ : $$2x^2y = (x+2)^2(y + 1), $$ provided that $(x+2)^2(y + 1)> 1000$.

PEN H Problems, 57
Tags: binomial coefficient , diophantine equation
Show that the equation ${n \choose k}=m^{l}$ has no integral solution with $l \ge 2$ and $4 \le k \le n-4$.

2019 Spain Mathematical Olympiad, 4
Tags: number theory , diophantine equation
Find all pairs of integers $(x,y)$ that satisfy the equation $3^4 2^3(x^2+y^2)=x^3y^3$

1967 Poland - Second Round, 4
Tags: diophantine , diophantine equation , number theory
Solve the equation in natural numbers $$ xy+yz+zx = xyz + 2. $$

2021 Austrian MO National Competition, 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)

2017 Saudi Arabia Pre-TST + Training Tests, 7
Tags: number theory , diophantine equation , diophantine
Find all pairs of integers $(x, y)$ such that $y^3 = 8x^6 + 2x^3 y -y^2$.

2017 India IMO Training Camp, 2
Tags: number theory , diophantine equation
Find all positive integers $p,q,r,s>1$ such that $$p!+q!+r!=2^s.$$

2009 Irish Math Olympiad, 3
Tags: diophantine equation , number theory , modular arithmetic
Find all pairs $(a,b)$ of positive integers such that $(ab)^2 - 4(a+b)$ is the square of an integer.

1999 French Mathematical Olympiad, Problem 2
Tags: diophantine equation , number theory
Find all natural numbers $n$ such that $$(n+3)^n=\sum_{k=3}^{n+2}k^n.$$

1998 German National Olympiad, 3
Tags: diophantine , diophantine equation , number theory
For each nonnegative integer $k$ find all nonnegative integers $x,y,z$ such that $x^2 +y^2 +z^2 = 8^k$