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

2022 Saudi Arabia JBMO TST, 1
Tags: number theory , diophantine equation
Find all pairs of positive prime numbers $(p, q)$ such that $$p^5 + p^3 + 2 = q^2 - q.$$

2015 QEDMO 14th, 3
Tags: number theory , diophantine , diophantine equation
Are there any rational numbers $x,y$ with $x^2 + y^2 = 2015$?

PEN H Problems, 78
Tags: diophantine equation
Let $x, y$, and $z$ be integers with $z>1$. Show that \[(x+1)^{2}+(x+2)^{2}+\cdots+(x+99)^{2}\neq y^{z}.\]

1992 Rioplatense Mathematical Olympiad, Level 3, 2
Tags: diophantine , diophantine equation , number theory
Determine the integers $0 \le a \le b \le c \le d$ such that: $$2^n= a^2 + b^2 + c^2 + d^2.$$

2015 District Olympiad, 2
Tags: diophantine equation , integer part , number theory
Determine the real numbers $ a,b, $ such that $$ [ax+by]+[bx+ay]=(a+b)\cdot [x+y],\quad\forall x,y\in\mathbb{R} , $$ where $ [t] $ is the greatest integer smaller than $ t. $

1993 Tournament Of Towns, (360) 3
Tags: number theory , perfect square , diophantine , diophantine equation
Positive integers $a$, $b$ and $c$ are positive integers with greatest common divisor equal to $1$ (i.e. they have no common divisors greater than $1$), and $$\frac{ab}{a-b}=c$$ Prove that $a -b$ is a perfect square. (SL Berlov)

2021 Mediterranean Mathematics Olympiad, 2
Tags: number theory , diophantine equation , diophantine
For every sequence $p_1<p_2<\cdots<p_8$ of eight prime numbers, determine the largest integer $N$ for which the following equation has no solution in positive integers $x_1,\ldots,x_8$: $$p_1\, p_2\, \cdots\, p_8 \left( \frac{x_1}{p_1}+ \frac{x_2}{p_2}+ ~\cdots~ +\frac{x_8}{p_8} \right) ~~=~~ N $$ [i]Proposed by Gerhard Woeginger, Austria[/i]

2020 Chile National Olympiad, 4
Tags: diophantine equation , number theory , diophantine , system of equations
Determine all three integers $(x, y, z)$ that are solutions of the system $$x + y -z = 6$$ $$x^3 + y^3 -z^3 = 414$$

2016 India PRMO, 2
Tags: algebra , factorial , equation , diophantine , diophantine equation , wilson
Find the number of integer solutions of the equation $x^{2016} + (2016! + 1!) x^{2015} + (2015! + 2!) x^{2014} + ... + (1! + 2016!) = 0$

2025 Kosovo EGMO Team Selection Test, P2
Tags: nt , diophantine equation , number theory
Find all natural numbers $m$ and $n$ such that $3^m+n!-1$ is the square of a natural number.

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 $10$.

2021 Middle European Mathematical Olympiad, 7
Tags: number theory , diophantine equation , easy number theorem
Find all pairs $(n, p)$ of positive integers such that $p$ is prime and \[ 1 + 2 + \cdots + n = 3 \cdot (1^2 + 2^2 + \cdot + p^2). \]

2013 Saudi Arabia BMO TST, 3
Tags: number theory , diophantine , diophantine equation
Find all positive integers $x, y, z$ such that $2^x + 21^y = z^2$

1989 Tournament Of Towns, (226) 4
Tags: number theory , diophantine , diophantine equation
Find the positive integer solutions of the equation $$ x+ \frac{1}{y+ \frac{1}{z}}= \frac{10}{7}$$ (G. Galperin)

2020 Czech-Austrian-Polish-Slovak Match, 5
Tags: number theory , diophantine equation
Let $n$ be a positive integer and let $d(n)$ denote the number of ordered pairs of positive integers $(x,y)$ such that $(x+1)^2-xy(2x-xy+2y)+(y+1)^2=n$. Find the smallest positive integer $n$ satisfying $d(n) = 61$. (Patrik Bak, Slovakia)

2009 IMAC Arhimede, 5
Tags: number theory , diophantine equation , diophantine , exponential equation , exponential
Find all natural numbers $x$ and $y$ such that $x^y-y^x=1$ .

2019 Moroccan TST, 3
Tags: number theory , diophantine equation
Find all couples $(x,y)$ over the positive integers such that: $7^x+x^4+47=y^2$

1985 Brazil National Olympiad, 4
Tags: integer , diophantine equation , number theory , trinomial
$a, b, c, d$ are integers. Show that $x^2 + ax + b = y^2 + cy + d$ has infinitely many integer solutions iff $a^2 - 4b = c^2 - 4d$.

2007 Puerto Rico Team Selection Test, 2
Tags: number theory , diophantine equation , diophantine , algebra , system of equations
Find the solutions of positive integers for the system $xy + x + y = 71$ and $x^2y + xy^2 = 880$.

1969 Swedish Mathematical Competition, 1
Tags: number theory , diophantine equation , diophantine
Find all integers m, n such that $m^3 = n^3 + n$.

1967 IMO Shortlist, 2
Tags: polynomial , diophantine equation , parametric equation , root , algebra
The equation \[x^5 + 5 \lambda x^4 - x^3 + (\lambda \alpha - 4)x^2 - (8 \lambda + 3)x + \lambda \alpha - 2 = 0\] is given. Determine $\alpha$ so that the given equation has exactly (i) one root or (ii) two roots, respectively, independent from $\lambda.$

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

2021 Taiwan TST Round 1, 3
Tags: diophantine equation , number theory
Find all triples $(x, y, z)$ of positive integers such that \[x^2 + 4^y = 5^z. \] [i]Proposed by Li4 and ltf0501[/i]

2013 JBMO Shortlist, 6
Tags: diophantine equation , integer , number theory
Solve in integers the system of equations: $$x^2-y^2=z$$ $$3xy+(x-y)z=z^2$$

2013 Bulgaria National Olympiad, 6
Tags: diophantine equation , number theory
Given $m\in\mathbb{N}$ and a prime number $p$, $p>m$, let \[M=\{n\in\mathbb{N}\mid m^2+n^2+p^2-2mn-2mp-2np \,\,\, \text{is a perfect square} \} \] Prove that $|M|$ does not depend on $p$. [i]Proposed by Aleksandar Ivanov[/i]