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

2021 New Zealand MO, 4
Tags: diophantine equation , diophantine , number theory
Find all triples $(x, p, n)$ of non-negative integers such that $p$ is prime and $2x(x + 5) = p^n + 3(x - 1)$.

II Soros Olympiad 1995 - 96 (Russia), 9.2
Tags: diophantine equation , diophantine , number theory
Find the integers $x, y, z$ for which $$\dfrac{1}{x+\dfrac{1}{y+\dfrac{1}{z}}}=\dfrac{7}{17}$$

1979 IMO Longlists, 44
Tags: diophantine equation , algebra , system of equations , cauchy-schwarz inequality
Determine all real numbers a for which there exists positive reals $x_{1}, \ldots, x_{5}$ which satisfy the relations $ \sum_{k=1}^{5} kx_{k}=a,$ $ \sum_{k=1}^{5} k^{3}x_{k}=a^{2},$ $ \sum_{k=1}^{5} k^{5}x_{k}=a^{3}.$

1984 Bulgaria National Olympiad, Problem 1
Tags: number theory , diophantine equation
Solve the equation $5^x7^y+4=3^z$ in nonnegative integers.

2018 Regional Olympiad of Mexico Northeast, 3
Tags: diophantine equation , number theory
Find the smallest natural number $n$ for which there exists a natural number $x$ such that $$(x+1)^3 + (x + 2)^3 + (x + 3)^3 + (x + 4)^3 = (x + n)^3.$$

2002 Junior Balkan Team Selection Tests - Romania, 1
Tags: number theory , diophantine equation , diophantine
Let $m,n > 1$ be integer numbers. Solve in positive integers $x^n+y^n = 2^m$.

2016 Hanoi Open Mathematics Competitions, 8
Tags: number theory , diophantine equation , diophantine
Find all positive integers $x,y,z$ such that $x^3 - (x + y + z)^2 = (y + z)^3 + 34$

2016 Czech-Polish-Slovak Junior Match, 6
Tags: diophantine equation , system of equations , diophantine , number theory
Let $k$ be a given positive integer. Find all triples of positive integers $a, b, c$, such that $a + b + c = 3k + 1$, $ab + bc + ca = 3k^2 + 2k$. Slovakia

2013 Iran MO (3rd Round), 4
Tags: pell equation , algebra , polynomial , vieta , number theory , diophantine equation
Prime $p=n^2 +1$ is given. Find the sets of solutions to the below equation: \[x^2 - (n^2 +1)y^2 = n^2.\] (25 points)

2014 Indonesia MO Shortlist, N3
Tags: diophantine equation , number theory , exponential equation
Find all pairs of natural numbers $(a, b)$ that fulfill $a^b=(a+b)^a$.

2021 Kazakhstan National Olympiad, 5
Tags: diophantine equation , algebra , number theory
Let $a$ be a positive integer. Prove that for any pair $(x,y)$ of integer solutions of equation $$x(y^2-2x^2)+x+y+a=0$$ we have: $$|x| \leqslant a+\sqrt{2a^2+2}$$

PEN H Problems, 71
Tags: diophantine equation , modular arithmetic
Let $n$ be a positive integer. Prove that the equation \[x+y+\frac{1}{x}+\frac{1}{y}=3n\] does not have solutions in positive rational numbers.

1997 All-Russian Olympiad Regional Round, 8.7
Tags: number theory , diophantine , diophantine equation
Find all pairs of prime numbers $p$ and $q$ such that $p^3-q^5 = (p+q)^2$.

2017 German National Olympiad, 6
Tags: diophantine equation , number theory
Prove that there exist infinitely many positive integers $m$ such that there exist $m$ consecutive perfect squares with sum $m^3$. Specify one solution with $m>1$.

2021 China Team Selection Test, 3
Tags: algebra , number theory , diophantine equation
Given positive integers $a,b,c$ which are pairwise coprime. Let $f(n)$ denotes the number of the non-negative integer solution $(x,y,z)$ to the equation $$ax+by+cz=n.$$ Prove that there exists constants $\alpha, \beta, \gamma \in \mathbb{R}$ such that for any non-negative integer $n$, $$|f(n)- \left( \alpha n^2+ \beta n + \gamma \right) | < \frac{1}{12} \left( a+b+c \right).$$

2021 New Zealand MO, 5
Tags: diophantine equation , number theory , diophantine
Find all pairs of integers $x, y$ such that $y^5 + 2xy = x^2 + 2y^4.$ .

1999 Belarusian National Olympiad, 6
Tags: diophantine equation , number theory
Solve in integer:$x^6+x^2y=y^3+2y^2$

2024 Israel TST, P1
Tags: number theory , diophantine equation , exponential equation
Solve in positive integers: \[x^{y^2+1}+y^{x^2+1}=2^z\]

2016 Postal Coaching, 5
Tags: number theory , diophantine equation
Find all nonnegative integers $k, n$ which satisfy $2^{2k+1} + 9\cdot 2^k + 5 = n^2.$

2017 Switzerland - Final Round, 7
Tags: number theory , perfect square , diophantine , diophantine equation
Let $n$ be a natural number such that there are exactly$ 2017$ distinct pairs of natural numbers $(a, b)$, which the equation $$\frac{1}{a}+\frac{1}{b}=\frac{1}{n}$$ fulfilld. Show that $n$ is a perfect square . Remark: $(7, 4) \ne (4, 7)$

2019 Romania National Olympiad, 4
Tags: number theory , diophantine equation , diophantine
Find the natural numbers $x, y, z$ that verify the equation: $$2^x + 3 \cdot 11^y =7^z$$

2001 Moldova National Olympiad, Problem 8
Tags: number theory , diophantine equation
Let $S$ be the set of positive integers $x$ for which there exist positive integers $y$ and $m$ such that $y^2-2^m=x^2$. (a) Find all of the elements of $S$. (b) Find all $x$ such that both $x$ and $x+1$ are in $S$.

2024 Bangladesh Mathematical Olympiad, P1
Tags: number theory , diophantine equation , modular arithmetic , prime number
Find all prime numbers $p$ and $q$ such that\[p^3-3^q=10.\] [i]Proposed by Md. Fuad Al Alam[/i]

2021 SG Originals, Q5
Tags: diophantine equation , number theory , algebra
Find all $a,b \in \mathbb{N}$ such that $$2049^ba^{2048}-2048^ab^{2049}=1.$$ [i]Proposed by fattypiggy123 and 61plus[/i]

2020 Azerbaijan Senior NMO, 2
Tags: perfect cube , diophantine equation , number theory
$a;b;c;d\in\mathbb{Z^+}$. Solve the equation: $$2^{a!}+2^{b!}+2^{c!}=d^3$$