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

2016 Hanoi Open Mathematics Competitions, 5

There are positive integers $x, y$ such that $3x^2 + x = 4y^2 + y$, and $(x - y)$ is equal to (A): $2013$ (B): $2014$ (C): $2015$ (D): $2016$ (E): None of the above.

2013 Hanoi Open Mathematics Competitions, 5

The number of integer solutions $x$ of the equation below $(12x -1)(6x - 1)(4x -1)(3x - 1) = 330$ is (A): $0$, (B): $1$, (C): $2$, (D): $3$, (E): None of the above.

1993 Tournament Of Towns, (360) 3

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)

2000 IMO Shortlist, 5

Prove that there exist infinitely many positive integers $ n$ such that $ p \equal{} nr,$ where $ p$ and $ r$ are respectively the semiperimeter and the inradius of a triangle with integer side lengths.

2019 South Africa National Olympiad, 6

Determine all pairs $(m, n)$ of non-negative integers that satisfy the equation $$ 20^m - 10m^2 + 1 = 19^n. $$

IV Soros Olympiad 1997 - 98 (Russia), 11.1

Solve the equation $xy =1997(x + y)$ in integers.

2025 Bangladesh Mathematical Olympiad, P4

Find all prime numbers $p, q$ such that$$p(p+1)(p^2+1) = q^2(q^2+q+1) + 2025.$$ [i]Proposed by Md. Fuad Al Alam[/i]

2001 Grosman Memorial Mathematical Olympiad, 6

(a) Find a pair of integers (x,y) such that $15x^2 +y^2 = 2^{2000}$ (b) Does there exist a pair of integers $(x,y)$ such that $15x^2 + y^2 = 2^{2000}$ and $x$ is odd?

2009 Iran Team Selection Test, 2

Let $ a$ be a fix natural number . Prove that the set of prime divisors of $ 2^{2^{n}} \plus{} a$ for $ n \equal{} 1,2,\cdots$ is infinite

2012 Dutch BxMO/EGMO TST, 3

Find all pairs of positive integers $(x,y) $ for which $x^3 + y^3 = 4(x^2y + xy^2 - 5) .$

1956 Poland - Second Round, 4

Prove that the equation $ 2x^2 - 215y^2 = 1 $ has no integer solutions.

1991 Irish Math Olympiad, 1

Problem. The sum of two consecutive squares can be a square; for instance $3^2 + 4^2 = 5^2$. (a) Prove that the sum of $m$ consecutive squares cannot be a square for $m \in \{3, 4, 5, 6\}$. (b) Find an example of eleven consecutive squares whose sum is a square. Can anyone help me with this? Thanks.

2014 Austria Beginners' Competition, 1

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

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

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

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

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

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

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) $x + y = a + b$.

2017 Harvard-MIT Mathematics Tournament, 5

Find the number of ordered triples of positive integers $(a, b, c)$ such that \[6a + 10b + 15c = 3000.\]

2003 All-Russian Olympiad Regional Round, 11.1

Find all prime $p$, for each of which there are such natural $ x$ and $y$ such that $p^x = y^3 + 1$.

PEN H Problems, 58

Solve in positive integers the equation $10^{a}+2^{b}-3^{c}=1997$.

2015 Costa Rica - Final Round, N1

Find all the values of $n \in N$ such that $n^2 = 2^n$.

2021 Nigerian MO Round 3, Problem 1

Find all triples of primes $(p, q, r)$ such that $p^q=2021+r^3$.