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: 436

2010 Hanoi Open Mathematics Competitions, 9
Tags: diophantine , number theory
Let $x,y$ be the positive integers such that $3x^2 +x = 4y^2 + y$. Prove that $x - y$ is a perfect (square).

2012 Cuba MO, 5
Tags: number theory , diophantine , diophantine equation
Find all pairs $(m, n)$ of positive integers such that $m^2 + n^2 =(m + 1)(n + 1).$

2019 Belarus Team Selection Test, 4.2
Tags: number theory , diophantine
Four positive integers $x,y,z$ and $t$ satisfy the relations \[ xy - zt = x + y = z + t. \] Is it possible that both $xy$ and $zt$ are perfect squares?

2019 Ecuador NMO (OMEC), 5
Tags: number theory , diophantine
Let $a, b, c$ be integers not all the same with $a, b, c\ge 4$ that satisfy $$4abc = (a + 3) (b + 3) (c + 3).$$ Find the numerical value of $a + b + c$.

1992 All Soviet Union Mathematical Olympiad, 580
Tags: inequalities , diophantine , algebra
If $a > b > c > d > 0$ are integers such that $ad = bc$, show that $$(a - d)^2 \ge 4d + 8$$

2018 Regional Olympiad of Mexico Center Zone, 5
Tags: diophantine equation , diophantine , number theory
Find all solutions of the equation $$p ^ 2 + q ^ 2 + 49r ^ 2 = 9k ^ 2-101$$ with $ p$, $q$ and $r$ positive prime numbers and $k$ a positive integer.

1976 Swedish Mathematical Competition, 6
Tags: diophantine , diophantine equation , number theory
Show that there are only finitely many integral solutions to \[ 3^m - 1 = 2^n \] and find them.

2019 Hanoi Open Mathematics Competitions, 11
Tags: number theory , diophantine , diophantine equation
Find all integers $x$ and $y$ satisfying the following equation $x^2 - 2xy + 5y^2 + 2x - 6y - 3 = 0$.

2016 Costa Rica - Final Round, N3
Tags: number theory , diophantine equation , diophantine
Find all nonnegative integers $a$ and $b$ that satisfy the equation $$3 \cdot 2^a + 1 = b^2.$$

2007 QEDMO 4th, 8
Tags: number theory , diophantine equation , diophantine
Show that there are no integers $x$ and $y$ satisfying $x^2 + 5 = y^3$. Daniel Harrer

2009 Postal Coaching, 2
Tags: diophantine , diophantine equation , number theory
Solve for prime numbers $p, q, r$ : $$\frac{p}{q} - \frac{4}{r + 1}= 1$$

2017 Swedish Mathematical Competition, 2
Tags: number theory , diophantine equation , diophantine
Let $p$ be a prime number. Find all pairs of coprime positive integers $(m,n)$ such that $$ \frac{p+m}{p+n}=\frac{m}{n}+\frac{1}{p^2}.$$

2011 Chile National Olympiad, 1
Tags: number theory , diophantine , diophantine equation
Find all the solutions $(a, b, c)$ in the natural numbers, verifying $1\le a \le b \le c$, of the equation$$\frac34=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}.$$

2011 Junior Balkan Team Selection Tests - Romania, 4
Tags: system of equations , diophantine equation , diophantine , number theory
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 $(x, y) \in Z \times Z$.

1994 Swedish Mathematical Competition, 4
Tags: number theory , diophantine , diophantine equation
Find all integers $m, n$ such that $2n^3 - m^3 = mn^2 + 11$.

2000 AMC 10, 22
Tags: diophantine , inequality
One morning each member of Angela's family drank an $ 8$-ounce mixture of coffee with milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Angela drank a quarter of the total amount of milk and a sixth of the total amount of coffee. How many people are in the family? $ \textbf{(A)}\ 3\qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 7$

2013 Costa Rica - Final Round, N1
Tags: number theory , diophantine equation , diophantine
Find all triples $(a, b, p)$ of positive integers, where $p$ is a prime number, such that $a^p - b^p = 2013$.

2010 Junior Balkan Team Selection Tests - Romania, 1
Tags: diophantine , number theory , prime
Determine the prime numbers $p, q, r$ with the property $\frac {1} {p} + \frac {1} {q} + \frac {1} {r} \ge 1$

1986 ITAMO, 6
Tags: diophantine , diophantine equation , number theory
Show that for any positive integer $n$ there exists an integer $m > 1$ such that $(\sqrt2-1)^n=\sqrt{m}-\sqrt{m-1}$.

2022-23 IOQM India, 4
Tags: algebra , diophantine
Starting with a positive integer $M$ written on the board , Alice plays the following game: in each move, if $x$ is the number on the board, she replaces it with $3x+2$.Similarly, starting with a positive integer $N$ written on the board, Bob plays the following game: in each move, if $x$ is the number on the board, he replaces it with $2x+27$.Given that Alice and Bob reach the same number after playing $4$ moves each, find the smallest value of $M+N$

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

2004 Junior Balkan Team Selection Tests - Romania, 3
Tags: number theory , diophantine equation , diophantine
Let $p, q, r$ be primes and let $n$ be a positive integer such that $p^n + q^n = r^2$. Prove that $n = 1$. Laurentiu Panaitopol

1973 Chisinau City MO, 70
Tags: diophantine , diophantine equation , number theory
The natural numbers $p, q$ satisfy the relation $p^p + q^q = p^q + q^p$. Prove that $p = q$.

2015 China Northern MO, 1
Tags: diophantine equation , diophantine , number theory
Find all integer solutions to the equation $$\frac{xyz}{w}+\frac{yzw}{x}+\frac{zwx}{y}+\frac{wxy}{z}=4$$

2011 NZMOC Camp Selection Problems, 6
Tags: diophantine equation , diophantine , number theory
Find all pairs of non-negative integers $m$ and $n$ that satisfy $$3 \cdot 2^m + 1 = n^2.$$