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

2004 Singapore MO Open, 2
Tags: number theory , diophantine , diophantine equation
Find the number of ordered pairs $(a, b)$ of integers, where $1 \le a, b \le 2004$, such that $x^2 + ax + b = 167 y$ has integer solutions in $x$ and $y$. Justify your answer.

2009 QEDMO 6th, 1
Tags: number theory , diophantine , diophantine equation
Solve $y^5 - x^2 = 4$ in integers numbers $x,y$.

2018 Hanoi Open Mathematics Competitions, 11
Tags: number theory , diophantine equation
Find all pairs of nonnegative integers $(x, y)$ for which $(xy + 2)^2 = x^2 + y^2 $.

2012 IFYM, Sozopol, 3
Tags: number theory , diophantine equation
Find all pairs of positive integers $(x,y) $ for which $x^3 + y^3 = 4(x^2y + xy^2 - 5) .$

JOM 2015 Shortlist, N4
Tags: number theory , diophantine equation
Determine all triplet of non-negative integers $ (x,y,z) $ satisfy $$ 2^x3^y+1=7^z $$

2014 Hanoi Open Mathematics Competitions, 11
Tags: number theory , diophantine equation , diophantine
Find all pairs of integers $(x,y)$ satisfying the following equality $8x^2y^2 + x^2 + y^2 = 10xy$

2015 District Olympiad, 2
Tags: number theory , diophantine equation , integer part
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. $

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

2022 USAMO, 4
Tags: number theory , diophantine equation , prime number
Find all pairs of primes $(p, q)$ for which $p-q$ and $pq-q$ are both perfect squares.

2020 Malaysia IMONST 1, 16
Tags: number theory , diophantine equation
Find the number of positive integer solutions $(a,b,c,d)$ to the equation \[(a^2+b^2)(c^2-d^2)=2020.\] Note: The solutions $(10,1,6,4)$ and $(1,10,6,4)$ are considered different.

2013 Junior Balkan Team Selection Tests - Romania, 2
Tags: diophantine equation , diophantine , number theory
Find all positive integers $x,y,z$ such that $7^x + 13^y = 8^z$

2011 Junior Balkan Team Selection Tests - Romania, 4
Tags: number theory , system of equations , diophantine equation , diophantine
Show that there is an infinite number of positive integers $t$ such that none of the equations $$ \begin{cases} x^2 + y^6 = t \\ x^2 + y^6 = t + 1 \\ x^2 - y^6 = t \\ x^2 - y^6 = t + 1 \end{cases}$$ has solutions $(x, y) \in Z \times Z$.

2017 Latvia Baltic Way TST, 14
Tags: diophantine , number theory , diophantine equation
Can you find three natural numbers $a, b, c$ whose greatest common divisor is $1$ and which satisfy the equality $$ab + bc + ac = (a + b -c)(b + c - a)(c + a - b) ?$$

2013 India IMO Training Camp, 1
Tags: number theory , modular arithmetic , diophantine equation
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 $10$.

1997 Brazil Team Selection Test, Problem 3
Tags: number theory , diophantine equation
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 $6k-1$.

1992 IMO Longlists, 36
Tags: diophantine equation , system of equations , algebra , polynomial
Find all rational solutions of \[a^2 + c^2 + 17(b^2 + d^2) = 21,\]\[ab + cd = 2.\]

2023 Bangladesh Mathematical Olympiad, P1
Tags: number theory , factorial , diophantine equation
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, $5! = 5\times 4\times 3\times 2\times 1 = 120$.

1964 Swedish Mathematical Competition, 2
Tags: diophantine , number theory , diophantine equation
Find all positive integers $m, n$ such that $n + (n+1) + (n+2) + ...+ (n+m) = 1000$.

1991 Romania Team Selection Test, 8
Tags: function , algebra , number theory , greatest common divisor , diophantine equation
Let $n, a, b$ be integers with $n \geq 2$ and $a \notin \{0, 1\}$ and let $u(x) = ax + b$ be the function defined on integers. Show that there are infinitely many functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that $f_n(x) = \underbrace{f(f(\cdots f}_{n}(x) \cdots )) = u(x)$ for all $x$. If $a = 1$, show that there is a $b$ for which there is no $f$ with $f_n(x) \equiv u(x)$.

PEN H Problems, 17
Tags: diophantine equation , vieta jumping
Find all positive integers $n$ for which the equation \[a+b+c+d=n \sqrt{abcd}\] has a solution in positive integers.

2002 Junior Balkan Team Selection Tests - Romania, 2
Tags: number theory , diophantine , diophantine equation , divisor
Let $k,n,p$ be positive integers such that $p$ is a prime number, $k < 1000$ and $\sqrt{k} = n\sqrt{p}$. a) Prove that if the equation $\sqrt{k + 100x} = (n + x)\sqrt{p}$ has a non-zero integer solution, then $p$ is a divisor of $10$. b) Find the number of all non-negative solutions of the above equation.

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

2002 Romania Team Selection Test, 2
Tags: number theory , polynomial , algebra , diophantine equation , induction
The sequence $ (a_n)$ is defined by: $ a_0\equal{}a_1\equal{}1$ and $ a_{n\plus{}1}\equal{}14a_n\minus{}a_{n\minus{}1}$ for all $ n\ge 1$. Prove that $ 2a_n\minus{}1$ is a perfect square for any $ n\ge 0$.

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

2015 India PRMO, 11
Tags: number theory , algebra , integer , diophantine equation
$11.$ Let $a,$ $b,$ and $c$ be real numbers such that $a-7b+8c=4.$ and $8a+4b-c=7.$ What is the value of $a^2-b^2+c^2 ?$