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: 397

1938 Eotvos Mathematical Competition, 1
Tags: number theory , Perfect Square
Prove that an integer $n$ can be expressed as the sum of two squares if and only if $2n$ can be expressed as the sum of two squares.

2018 Greece Junior Math Olympiad, 3
Tags: number theory , Perfect Square
Let $a$ and $b$ be positive integers with $b$ odd, such that the number $$\frac{(a+b)^2+4a}{ab}$$ is an integer. Prove that $a$ is a perfect square.

1999 Switzerland Team Selection Test, 6
Tags: divisible , Perfect Square , number theory
Prove that if $m$ and $n$ are positive integers such that $m^2 + n^2 - m$ is divisible by $2mn$, then $m$ is a perfect square.

2015 Caucasus Mathematical Olympiad, 5
Tags: number theory , Perfect Square , Sides of a triangle
Are there natural $a, b >1000$ , such that for any $c$ that is a perfect square, the three numbers $a, b$ and $c$ are not the lengths of the sides of a triangle?

1994 Nordic, 4
Tags: number theory , Perfect Square , Integer , Pell equations
Determine all positive integers $n < 200$, such that $n^2 + (n+ 1)^2$ is the square of an integer.

1996 Dutch Mathematical Olympiad, 5
Tags: number theory , greatest common divisor , Perfect Square
For the positive integers $x , y$ and $z$ apply $\frac{1}{x}+\frac{1}{y}=\frac{1}{z}$ . Prove that if the three numbers $x , y,$ and $z$ have no common divisor greater than $1$, $x + y$ is the square of an integer.

2001 Polish MO Finals, 1
Tags: modular arithmetic , function , number theory , Perfect Square
Assume that $a,b$ are integers and $n$ is a natural number. $2^na+b$ is a perfect square for every $n$.Prove that $a=0$.

1984 All Soviet Union Mathematical Olympiad, 387
Tags: Perfect Square , number theory
The $x$ and $y$ figures satisfy a condition: for every $n\ge1$ the number $$xx...x6yy...y4$$ ($n$ times $x$ and $n$ times $y$) is a perfect square. Find all possible $x$ and $y$.

2005 Slovenia Team Selection Test, 3
Tags: Perfect Square , number theory
Find all pairs $(m, n)$ of natural numbers for which the numbers $m^2 - 4n$ and $n^2 - 4m$ are both perfect squares.

ICMC 6, 4
Tags: ICMC , college contests , number theory , sum of divisors , Perfect Square
Do there exist infinitely many positive integers $m$ such that the sum of the positive divisors of $m$ (including $m$ itself) is a perfect square? [i]Proposed by Dylan Toh[/i]

2015 QEDMO 14th, 10
Tags: number theory , Perfect Square
Find all prime numbers $p$ for which $p^3- p + 1$ is a perfect square .

OIFMAT III 2013, 4
Tags: Perfect Squares , Perfect Square , Perfect power , number theory
Show that there exists a set of infinite positive integers such that the sum of an arbitrary finite subset of these is never a perfect square. What happens if we change the condition from not being a perfect square to not being a perfect power?

1996 Bundeswettbewerb Mathematik, 4
Tags: number theory , Perfect Square
Find all natural numbers $n$ for which $n2^{n-1} +1$ is a perfect square.

1994 Greece National Olympiad, 1
Tags: number theory , Perfect Squares , Perfect Square
Prove that number $2(1991m^2+1993mn+1995n^2)$ where $m,n$ are poitive integers, cannot be a square of an integer.

2006 Thailand Mathematical Olympiad, 1
Tags: number theory , consecutive , Perfect Square
Show that the product of three consecutive positive integers is never a perfect square.

2007 Thailand Mathematical Olympiad, 4
Tags: number theory , Perfect Square
Find all primes $p$ such that $\frac{2^{p-1}-1}{p}$ is a perfect square.

2003 Croatia Team Selection Test, 1
Tags: Perfect Square , number theory
Find all pairs $(m, n)$ of natural numbers for which the numbers $m^2 - 4n$ and $n^2 - 4m$ are both perfect squares.

1986 All Soviet Union Mathematical Olympiad, 423
Tags: Perfect Square , table , number theory , Sum , combinatorics
Prove that the rectangle $m\times n$ table can be filled with exact squares so, that the sums in the rows and the sums in the columns will be exact squares also.

2013 Kyiv Mathematical Festival, 2
Tags: number theory , representation , Sum , Perfect Square
For which positive integers $n \ge 2$ it is possible to represent the number $n^2$ as a sum of n distinct positive integers not exceeding $\frac{3n}{2}$ ?

2011 District Olympiad, 2
Tags: number theory , Perfect Square
a) Show that $m^2- m +1$ is an element of the set $\{n^2 + n +1 | n \in N\}$, for any positive integer $ m$. b) Let $p$ be a perfect square, $p> 1$. Prove that there exists positive integers $r$ and $q$ such that $$p^2 + p +1=(r^2 + r + 1)(q^2 + q + 1).$$

OIFMAT III 2013, 1
Tags: number theory , Digits , Perfect Square
Find all four-digit perfect squares such that: $\bullet$ All your figures are less than $9$. $\bullet$ By increasing each of its digits by one unit, the resulting number is again a perfect square.

1988 IMO Shortlist, 9
Tags: number theory , Divisibility , Diophantine equation , Perfect Square , Vieta Jumping , IMO , IMO 1988
Let $ a$ and $ b$ be two positive integers such that $ a \cdot b \plus{} 1$ divides $ a^{2} \plus{} b^{2}$. Show that $ \frac {a^{2} \plus{} b^{2}}{a \cdot b \plus{} 1}$ is a perfect square.

2021 Malaysia IMONST 1, 15
Tags: Perfect Square , Perfect Squares , number theory
Find the sum of all integers $n$ with this property: both $n$ and $n + 2021$ are perfect squares.

1999 Spain Mathematical Olympiad, 2
Tags: number theory , Perfect Square , Integer sequence
Prove that there exists a sequence of positive integers $a_1,a_2,a_3, ...$ such that $a_1^2+a_2^2+...+a_n^2$ is a perfect square for all positive integers $n$.

1981 Swedish Mathematical Competition, 6
Tags: number theory , Perfect Square , power of 2 , triangle inequality
Show that there are infinitely many triangles with side lengths $a$, $b$, $c$, where $a$ is a prime, $b$ is a power of $2$ and $c$ is the square of an odd integer.