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

2014 Contests, 1
Tags: diophantine equation , diophantine , number theory
Determine all triples $(a,b,c)$, where $a, b$, and $c$ are positive integers that satisfy $a \le b \le c$ and $abc = 2(a + b + c)$.

2009 Postal Coaching, 2
Tags: diophantine , diophantine equation , number theory
Determine, with proof, all the integer solutions of the equation $x^3 + 2y^3 + 4z^3 - 6xyz = 1$.

2019 Azerbaijan Junior NMO, 5
Tags: diophantine equation , number theory
Find all $x;y\in\mathbb{Z}$ satisfying the following condition: $$x^3=y^4+9x^2$$

1967 IMO Shortlist, 4
Tags: modular arithmetic , number theory , perfect cube , diophantine equation
Does there exist an integer such that its cube is equal to $3n^2 + 3n + 7,$ where $n$ is an integer.

2015 IMAR Test, 1
Tags: diophantine equation , ro
Determine all positive integers expressible, for every integer $ n \geq 3 $, in the form \begin{align*} \frac{(a_1 + 1)(a_2 + 1) \ldots (a_n + 1) - 1}{a_1a_2 \ldots a_n}, \end{align*} where $ a_1, a_2, \ldots, a_n $ are pairwise distinct positive integers.

2014 Belarus Team Selection Test, 4
Tags: number theory , diophantine , diophantine equation
Find all integers $a$ and $b$ satisfying the equality $3^a - 5^b = 2$. (I. Gorodnin)

2015 Bosnia and Herzegovina Junior BMO TST, 1
Tags: number theory , equation , algebra , diophantine equation
Solve equation $x(x+1) = y(y+4)$ where $x$, $y$ are positive integers

2023 Brazil National Olympiad, 3
Tags: diophantine equation , system of equations , number theory
Let $n$ be a positive integer. Show that there are integers $x_1, x_2, \ldots , x_n$, [i]not all equal[/i], satisfying $$\begin{cases} x_1^2+x_2+x_3+\ldots+x_n=0 \\ x_1+x_2^2+x_3+\ldots+x_n=0 \\ x_1+x_2+x_3^2+\ldots+x_n=0 \\ \vdots \\ x_1+x_2+x_3+\ldots+x_n^2=0 \end{cases}$$ if, and only if, $2n-1$ is not prime.

2023 Bulgaria JBMO TST, 3
Tags: number theory , prime number , diophantine equation
Find all natural numbers $a$, $b$, $c$ and prime numbers $p$ and $q$, such that: $\blacksquare$ $4\nmid c$ $\blacksquare$ $p\not\equiv 11\pmod{16}$ $\blacksquare$ $p^aq^b-1=(p+4)^c$

2008 Regional Olympiad of Mexico Center Zone, 1
Tags: diophantine equation , diophantine , number theory
Find all pairs of integers $ a, b $ that satisfy $a ^2-3a = b ^3-2$.

2010 Federal Competition For Advanced Students, P2, 2
Tags: number theory , diophantine equation , diophantine
Determine all triples $(x, y, z)$ of positive integers $x > y > z > 0$, such that $x^2 = y \cdot 2^z + 1$

1967 IMO Longlists, 43
Tags: algebra , polynomial , diophantine equation , parametric equation , root
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 $\lambda.$

1999 Swedish Mathematical Competition, 3
Tags: number theory , diophantine equation , diophantine
Find non-negative integers $a, b, c, d$ such that $5^a + 6^b + 7^c + 11^d = 1999$.

1997 Denmark MO - Mohr Contest, 4
Tags: number theory , diophantine , diophantine equation
Find all pairs $x,y$ of natural numbers that satisfy the equation $$x^2-xy+2x-3y=1997$$

2020 OMpD, 1
Tags: number theory , diophantine equation , exponential equation , polynomial , algebra
Determine all pairs of positive integers $(x, y)$ such that: $$x^4 - 6x^2 + 1 = 7\cdot 2^y$$

2013 Danube Mathematical Competition, 3
Tags: diophantine equation , diophantine , exponential equation , exponential , number theory
Determine the natural numbers $m,n$ such as $85^m-n^4=4$

2010 Saudi Arabia BMO TST, 1
Tags: diophantine , diophantine equation , number theory
Find all triples $(x,y,z)$ of positive integers such that $3^x + 4^y = 5^z$.

2007 Nordic, 1
Tags: diophantine equation , number theory
Find a solution to the equation $x^2-2x-2007y^2=0$ in positive integers.

1997 Estonia National Olympiad, 1
Tags: diophantine , diophantine equation , number theory
Find: a) Any quadruple of positive integers $(a, k, l, m)$ such that $a^k = a^l + a^m,$ b) Any quintuple of positive integers $(a, k, l, m, n)$ for which $a^k = a^l + a^m+a^n$

1971 IMO Longlists, 14
Tags: geometry , 3d geometry , diophantine equation , number theory
Note that $8^3 - 7^3 = 169 = 13^2$ and $13 = 2^2 + 3^2.$ Prove that if the difference between two consecutive cubes is a square, then it is the square of the sum of two consecutive squares.

1975 Chisinau City MO, 92
Tags: number theory , diophantine equation , diophantine
Solve in natural numbers the equation $x^2-y^2=105$.

1982 IMO Shortlist, 16
Tags: number theory , equation , diophantine equation , divisibility
Prove that if $n$ is a positive integer such that the equation \[ x^3-3xy^2+y^3=n \] has a solution in integers $x,y$, then it has at least three such solutions. Show that the equation has no solutions in integers for $n=2891$.

PEN H Problems, 4
Tags: diophantine equation , ellipse , parameterization , greatest common divisor , conic
Find all pairs $(x, y)$ of positive rational numbers such that $x^{2}+3y^{2}=1$.

2015 Saudi Arabia Pre-TST, 1.3
Tags: diophantine equation , diophantine , number theory
Find all integer solutions of the equation $x^2y^5 - 2^x5^y = 2015 + 4xy$. (Malik Talbi)

2012 Vietnam Team Selection Test, 1
Tags: algebra , polynomial , diophantine equation , number theory
Consider the sequence $(x_n)_{n\ge 1}$ where $x_1=1,x_2=2011$ and $x_{n+2}=4022x_{n+1}-x_n$ for all $n\in\mathbb{N}$. Prove that $\frac{x_{2012}+1}{2012}$ is a perfect square.