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

1956 Poland - Second Round, 4
Tags: diophantine , diophantine equation , number theory
Prove that the equation $ 2x^2 - 215y^2 = 1 $ has no integer solutions.

2014 Austria Beginners' Competition, 1
Tags: diophantine , diophantine equation , number theory
Determine all solutions of the diophantine equation $a^2 = b \cdot (b + 7)$ in integers $a\ge 0$ and $b \ge 0$. (W. Janous, Innsbruck)

2010 Regional Olympiad of Mexico Center Zone, 5
Tags: diophantine , diophantine equation , number theory
Find all integer solutions $(p, q, r)$ of the equation $r + p ^ 4 = q ^ 4$ with the following conditions: $\bullet$ $r$ is a positive integer with exactly $8$ positive divisors. $\bullet$ $p$ and $q$ are prime numbers.

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

2004 Junior Balkan Team Selection Tests - Romania, 3
Tags: diophantine , diophantine equation , number theory
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

1991 Tournament Of Towns, (291) 1
Tags: number theory , diophantine , diophantine equation
Find all natural numbers $n$, and all integers $x,y$ ($x\ne y$) for which the following equation is satisfied: $$x + x^2 + x^4 + ...+ x^{2^n} = y + y^2 + y^4 + ... + y^{2^n} .$$

2005 iTest, 16
Tags: number theory , diophantine , diophantine equation
How many distinct integral solutions of the form $(x, y)$ exist to the equation$ 21x + 22y = 43$ such that $1 < x < 11$ and $y < 22$?

1982 Austrian-Polish Competition, 7
Tags: diophantine , diophantine equation , system of equations , number theory
Find the triple of positive integers $(x,y,z)$ with $z$ least possible for which there are positive integers $a, b, c, d$ with the following properties: (i) $x^y = a^b = c^d$ and $x > a > c$ (ii) $z = ab = cd$ (iii) $x + y = a + b$.

2003 All-Russian Olympiad Regional Round, 11.1
Tags: diophantine , diophantine equation , number theory
Find all prime $p$, for each of which there are such natural $ x$ and $y$ such that $p^x = y^3 + 1$.

2015 Costa Rica - Final Round, N1
Tags: diophantine , diophantine equation , number theory
Find all the values of $n \in N$ such that $n^2 = 2^n$.

2019 Austrian Junior Regional Competition, 1
Tags: number theory , diophantine equation , diophantine
Let $x$ and $y$ be integers with $x + y \ne 0$. Find all pairs $(x, y)$ such that $$\frac{x^2 + y^2}{x + y}= 10.$$ (Walther Janous)

1969 Dutch Mathematical Olympiad, 2
Tags: number theory , diophantine , diophantine equation
Prove that for all $n \in N$, $x^2 + y^2 = z^n$ has solutions with $x,y,z \in N$.

2022 Czech-Polish-Slovak Junior Match, 4
Tags: number theory , diophantine equation , diophantine
Find all triples $(a, b, c)$ of integers that satisfy the equations $ a + b = c$ and $a^2 + b^3 = c^2$

2017 Saudi Arabia JBMO TST, 3
Tags: diophantine , diophantine equation , number theory
Find all pairs of primes $(p, q)$ such that $p^3 - q^5 = (p + q)^2$ .

2013 District Olympiad, 1
Tags: diophantine equation , number theory , diophantine
Find all triples of integers $(x, y, z)$ such that $$x^2 + y^2 + z^2 = 16(x + y + z).$$

2002 Junior Balkan Team Selection Tests - Romania, 1
Tags: diophantine , number theory , diophantine equation
Let $m,n > 1$ be integer numbers. Solve in positive integers $x^n+y^n = 2^m$.

1967 All Soviet Union Mathematical Olympiad, 089
Tags: diophantine , number theory
Find all the integers $x,y$ satisfying equation $x^2+x=y^4+y^3+y^2+y$.

2009 May Olympiad, 2
Tags: diophantine , diophantine equation , number theory
Find prime numbers $p , q , r$ such that $p+q^2+r^3=200$. Give all the possibilities. Remember that the number $1$ is not prime.

1984 Tournament Of Towns, (069) T3
Tags: algebra , number theory , diophantine equation , equation , diophantine
Find all solutions of $2^n + 7 = x^2$ in which n and x are both integers . Prove that there are no other solutions.

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

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$

1999 Estonia National Olympiad, 1
Tags: number theory , diophantine , diophantine equation
Find all pairs of integers ($a, b$) such that $a^2 + b = b^{1999}$ .

1954 Moscow Mathematical Olympiad, 262
Tags: diophantine , number theory , diophantine equation
Are there integers $m$ and $n$ such that $m^2 + 1954 = n^2$?

2002 Dutch Mathematical Olympiad, 2
Tags: diophantine , diophantine equation , number theory
Determine all triplets $(x, y, z)$ of positive integers with $x \le y \le z$ that satisfy $\left(1+\frac1x \right)\left(1+\frac1y \right)\left(1+\frac1z \right) = 3$

2010 China Northern MO, 3
Tags: diophantine , number theory , diophantine equation
Find all positive integer triples $(x, y, z)$ such that $1 + 2^x \cdot 3^y=5^z$ is true.