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

2009 District Olympiad, 4
Tags: number theory , diophantine equation
Positive integer numbers a and b satisfy $(a^2- 9b^2)^2 - 33b = 1$. a) Prove $|a -3b|\ge 1$. b) Find all pairs of positive integers $(a, b)$ satisfying the equality.

2023 Bangladesh Mathematical Olympiad, P1
Tags: diophantine equation , factorial , number theory
Find all possible non-negative integer solution $(x,y)$ of the following equation- $$x! + 2^y =(x+1)!$$ Note: $x!=x \cdot (x-1)!$ and $0!=1$. For example, $5! = 5\times 4\times 3\times 2\times 1 = 120$.

1980 IMO Shortlist, 3
Tags: number theory , equation , algebra , diophantine equation
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.

2011 India IMO Training Camp, 1
Tags: 3d geometry , diophantine equation , number theory , geometry
Find all positive integer $n$ satisfying the conditions $a)n^2=(a+1)^3-a^3$ $b)2n+119$ is a perfect square.

2010 Abels Math Contest (Norwegian MO) Final, 4a
Tags: diophantine equation , diophantine , number theory
Find all positive integers $k$ and $\ell$ such that $k^2 -\ell^2 = 1005$.

VI Soros Olympiad 1999 - 2000 (Russia), 9.1
Tags: number theory , diophantine equation
Prove that there is no natural number $k$ such that $k^{1999} - k^{1998} = 2k + 2$.

2014 Contests, 3
Tags: diophantine equation , number theory
Determine whether there exist an infinite number of positive integers $x,y $ satisfying the condition: $x^2+y \mid x+y^2.$ Please prove it.

2013 India IMO Training Camp, 1
Tags: modular arithmetic , diophantine equation , number theory
A positive integer $a$ is called a [i]double number[/i] if it has an even number of digits (in base 10) and its base 10 representation has the form $a = a_1a_2 \cdots a_k a_1 a_2 \cdots a_k$ with $0 \le a_i \le 9$ for $1 \le i \le k$, and $a_1 \ne 0$. For example, $283283$ is a double number. Determine whether or not there are infinitely many double numbers $a$ such that $a + 1$ is a square and $a + 1$ is not a power of $10$.

2010 NZMOC Camp Selection Problems, 3
Tags: diophantine , diophantine equation , number theory
Let $p$ be a prime number. Find all pairs $(x, y)$ of positive integers such that $x^3 + y^3 - 3xy = p -1$.

PEN H Problems, 30
Tags: diophantine equation
Let $a$, $b$, $c$ be given integers, $a>0$, $ac-b^2=p$ a squarefree positive integer. Let $M(n)$ denote the number of pairs of integers $(x, y)$ for which $ax^2 +bxy+cy^2=n$. Prove that $M(n)$ is finite and $M(n)=M(p^{k} \cdot n)$ for every integer $k \ge 0$.

2021 Israel TST, 4
Tags: diophantine equation , number theory
Let $r$ be a positive integer and let $a_r$ be the number of solutions to the equation $3^x-2^y=r$ ,such that $0\leq x,y\leq 5781$ are integers. What is the maximal value of $a_r$?

PEN H Problems, 51
Tags: diophantine equation , modular arithmetic
Prove that the product of five consecutive positive integers is never a perfect square.

2022 Turkey MO (2nd round), 4
Tags: diophantine equation , number theory
For which real numbers $a$, there exist pairwise different real numbers $x, y, z$ satisfying $$\frac{x^3+a}{y+z}=\frac{y^3+a}{x+z}=\frac{z^3+a}{x+y}= -3.$$

2005 USAMO, 2
Tags: modular arithmetic , diophantine equation , algebra
Prove that the system \begin{align*} x^6+x^3+x^3y+y & = 147^{157} \\ x^3+x^3y+y^2+y+z^9 & = 157^{147} \end{align*} has no solutions in integers $x$, $y$, and $z$.

PEN H Problems, 63
Tags: diophantine equation , zsigmondy therom
Show that $\vert 12^m -5^n\vert \ge 7$ for all $m, n \in \mathbb{N}$.

2011 Junior Balkan Team Selection Tests - Moldova, 6
Tags: number theory , diophantine , diophantine equation
Find the sum of the numbers written with two digits $\overline{ab}$ for which the equation $3^{x + y} =3^x + 3^y + \overline{ab}$ has at least one solution $(x, y)$ in natural numbers.

1984 IMO Longlists, 12
Tags: number theory , equation , polynomial , diophantine equation
Let $n$ be a positive integer and $a_1, a_2, \dots , a_{2n}$ mutually distinct integers. Find all integers $x$ satisfying \[(x - a_1) \cdot (x - a_2) \cdots (x - a_{2n}) = (-1)^n(n!)^2.\]

2017 QEDMO 15th, 6
Tags: number theory , diophantine , diophantine equation
Find all integers $x,y$ satisfy the $x^3 + y^3 = 3xy$.

2021 Junior Macedonian Mathematical Olympiad, Problem 3
Tags: number theory , diophantine equation
Find all positive integers $n$ and prime numbers $p$ such that $$17^n \cdot 2^{n^2} - p =(2^{n^2+3}+2^{n^2}-1) \cdot n^2.$$ [i]Authored by Nikola Velov[/i]

2023 Costa Rica - Final Round, 3.6
Tags: diophantine equation , number theory , digit
Given a positive integer $N$, define $u(N)$ as the number obtained by making the ones digit the left-most digit of $N$, that is, taking the last, right-most digit (the ones digit) and moving it leftwards through the digits of $N$ until it becomes the first (left-most) digit; for example, $u(2023) = 3202$. [b](1)[/b] Find a $6$-digit positive integer $N$ such that \[\frac{u(N)}{N} = \frac{23}{35}.\] [b](2)[/b] Prove that there is no positive integer $N$ with less than $6$ digits such that \[\frac{u(N)}{N} = \frac{23}{35}.\]

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 $6k-1$.

2015 District Olympiad, 3
Tags: diophantine equation , algebra
Find $ \#\left\{ (x,y)\in\mathbb{N}^2\bigg| \frac{1}{\sqrt{x}} -\frac{1}{\sqrt{y}} =\frac{1}{2016}\right\} , $ where $ \# A $ is the cardinal of $ A . $

2012 Mathcenter Contest + Longlist, 1
Tags: number theory , diophantine equation , diophantine
Prove without using modulo that there are no integers $a,b,c$ such that $$a^2+b^2-8c = 6$$ [i](Metamorphosis)[/i]

1975 Bulgaria National Olympiad, Problem 1
Tags: diophantine equation , number theory
Find all pairs of natural numbers $(m,n)$ bigger than $1$ for which $2^m+3^n$ is the square of whole number. [i]I. Tonov[/i]

2015 Dutch IMO TST, 2
Tags: diophantine equation , system of equations , diophantine , number theory
Determine all positive integers $n$ for which there exist positive integers $a_1,a_2, ..., a_n$ with $a_1 + 2a_2 + 3a_3 +... + na_n = 6n$ and $\frac{1}{a_1}+\frac{2}{a_2}+\frac{3}{a_3}+ ... +\frac{n}{a_n}= 2 + \frac1n$