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: 521

2017 Switzerland - Final Round, 7
Tags: perfect square , diophantine equation , diophantine , number theory
Let $n$ be a natural number such that there are exactly$ 2017$ distinct pairs of natural numbers $(a, b)$, which the equation $$\frac{1}{a}+\frac{1}{b}=\frac{1}{n}$$ fulfilld. Show that $n$ is a perfect square . Remark: $(7, 4) \ne (4, 7)$

1991 Bundeswettbewerb Mathematik, 1
Tags: diophantine , diophantine equation , perfect square , number theory
Determine all solutions of the equation $4^x + 4^y + 4^z = u^2$ for integers $x,y,z$ and $u$.

2022 Czech and Slovak Olympiad III A, 5
Tags: perfect square , number theory
Find all integers $n$ such that $2^n + n^2$ is a square of an integer. [i](Tomas Jurik )[/i]

2007 Thailand Mathematical Olympiad, 17
Tags: perfect square , number theory
Compute the product of positive integers $n$ such that $n^2 + 59n + 881$ is a perfect square.

2019 Swedish Mathematical Competition, 6
Tags: sequence , perfect square , number theory
Is there an infinite sequence of positive integers $\{a_n\}_{n = 1}^{\infty}$ which contains each positive integer exactly once and is such that the number $a_n + a_{n + 1} $ is a perfect square for each $n$?

1962 Czech and Slovak Olympiad III A, 1
Tags: quadratic polynomial , perfect square , prime , algebra
Determine all integers $x$ such that $2x^2-x-36$ is a perfect square of a prime.

1988 Bundeswettbewerb Mathematik, 1
Tags: perfect square , number theory
For the natural numbers $x$ and $y$, $2x^2 + x = 3y^2 + y$ . Prove that then $x-y$, $2x + 2y + 1$ and $3x + 3y + 1$ are perfect squares.

1974 Putnam, A3
Tags: perfect square , prime number , number theory
A well-known theorem asserts that a prime $p > 2$ can be written as the sum of two perfect squares ($p = m^2 +n^2$ , with $m$ and $n$ integers) if and only if $p \equiv 1$ (mod $4$). Assuming this result, find which primes $p > 2$ can be written in each of the following forms, using integers $x$ and $y$: a) $x^2 +16y^2, $ b) $4x^2 +4xy+ 5y^2.$

2010 Chile National Olympiad, 1
Tags: perfect square , number theory
The integers $a, b$ satisfy the following identity $$2a^2 + a = 3b^2 + b.$$ Prove that $a- b$, $2a + 2b + 1$, and $3a + 3b + 1$ are perfect squares.

2008 Dutch IMO TST, 4
Tags: perfect square , radical , number theory
Let $n$ be positive integer such that $\sqrt{1 + 12n^2}$ is an integer. Prove that $2 + 2\sqrt{1 + 12n^2}$ is the square of an integer.

2020 Taiwan TST Round 2, 2
Tags: ceiling function , perfect square , number theory
Let $a$ and $b$ be two positive integers. Prove that the integer \[a^2+\left\lceil\frac{4a^2}b\right\rceil\] is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to $z$.) [i]Russia[/i]

1995 Mexico National Olympiad, 4
Tags: perfect square , number theory
Find $26$ elements of $\{1, 2, 3, ... , 40\}$ such that the product of two of them is never a square. Show that one cannot find $27$ such elements.

2021 Bangladeshi National Mathematical Olympiad, 7
Tags: perfect cube , divisor , perfect square , function , number theory
For a positive integer $n$, let $s(n)$ and $c(n)$ be the number of divisors of $n$ that are perfect squares and perfect cubes respectively. A positive integer $n$ is called fair if $s(n)=c(n)>1$. Find the number of fair integers less than $100$.

2005 Bosnia and Herzegovina Junior BMO TST, 2
Tags: perfect square , integer , number theory
Let n be a positive integer. Prove the following statement: ”If $2 + 2\sqrt{1 + 28n^2}$ is an integer, then it is the square of an integer.”

2025 Bulgarian Spring Mathematical Competition, 10.3
Tags: table , perfect square , algebra
In the cell $(i,j)$ of a table $n\times n$ is written the number $(i-1)n + j$. Determine all positive integers $n$ such that there are exactly $2025$ rows not containing a perfect square.

2010 Saudi Arabia Pre-TST, 3.2
Tags: perfect square , divisor , number theory
Prove that among any nine divisors of $30^{2010}$ there are two whose product is a perfect square.

2021 Czech-Polish-Slovak Junior Match, 4
Tags: perfect square , number theory
Find the smallest positive integer $n$ with the property that in the set $\{70, 71, 72,... 70 + n\}$ you can choose two different numbers whose product is the square of an integer.

2019 Saudi Arabia JBMO TST, 4
Tags: perfect square , number theory
Let $p$ be a prime number. Show that $7^p+3p-4$ is not a perfect square.

2010 Contests, 3
Tags: algebra , perfect square , function , number theory
Find all functions $g:\mathbb{N}\rightarrow\mathbb{N}$ such that \[\left(g(m)+n\right)\left(g(n)+m\right)\] is a perfect square for all $m,n\in\mathbb{N}.$ [i]Proposed by Gabriel Carroll, USA[/i]

2009 Abels Math Contest (Norwegian MO) Final, 1b
Tags: perfect cube , perfect square , number theory
Show that the sum of three consecutive perfect cubes can always be written as the difference between two perfect squares.

2004 Mexico National Olympiad, 1
Tags: prime , perfect square , number theory
Find all the prime number $p, q$ and r with $p < q < r$, such that $25pq + r = 2004$ and $pqr + 1 $ is a perfect square.

1999 All-Russian Olympiad Regional Round, 10.5
Tags: perfect square , number theory
Are there $10$ different integers such that all the sums made up of $9$ of them are perfect squares?

2003 Belarusian National Olympiad, 2
Tags: perfect square , polynomial , algebra
Let $P(x) =(x+1)^p (x-3)^q=x^n+a_1x^{n-1}+a_2x^{n-2}+...+a_{n-1}x+a_n$ where $p$ and $q$ are positive integers a) Given $a_1=a_2$, prove that $3n$ is a perfect square. b) Prove that there exist infinitely many pairs $(p, q)$ of positive integers p and q such that the equality $a_1=a_2$ is valid for the polynomial $P(x)$. (D. Bazylev)

1979 IMO Shortlist, 23
Tags: perfect square , number theory
Find all natural numbers $n$ for which $2^8 +2^{11} +2^n$ is a perfect square.

2020 Olympic Revenge, 4
Tags: additive combinatorics , perfect square , number theory , combinatorics
Let $n$ be a positive integer and $A$ a set of integers such that the set $\{x = a + b\ |\ a, b \in A\}$ contains $\{1^2, 2^2, \dots, n^2\}$. Prove that there is a positive integer $N$ such that if $n \ge N$, then $|A| > n^{0.666}$.