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

1966 Poland - Second Round, 4
Tags: number theory , perfect square
Prove that if the natural numbers $ a $ and $ b $ satisfy the equation $ a^2+a = 3b^2 $, then the number $ a+1 $ is the square of an integer.

2020 Greece Team Selection Test, 4
Tags: ceiling function , number theory , perfect square
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]

1969 IMO Longlists, 62
Tags: number theory , perfect square
Which natural numbers can be expressed as the difference of squares of two integers?

1995 Tournament Of Towns, (475) 3
Tags: number theory , perfect square
The first digit of a $6$-digit number is $5$. Is it true that it is always possible to write $6$ more digits to the right of this number so that the resulting $12$-digit number is a perfect square? (A Tolpygo)

2000 IMO Shortlist, 6
Tags: number theory , additive number theory , sum of squares , perfect square
Show that the set of positive integers that cannot be represented as a sum of distinct perfect squares is finite.

2020 Peru Cono Sur TST., P4
Tags: number theory , phi function , euler s phi function , perfect square
Find all odd integers $n$ for which $\frac{2^{\phi (n)}-1}{n}$ is a perfect square.

2007 Cuba MO, 2
Tags: number theory , perfect square
Find three different positive integers whose sum is minimum than meet the condition that the sum of each pair of them is a perfect square.

1987 Mexico National Olympiad, 4
Tags: number theory , product , perfect square , number of divisors
Calculate the product of all positive integers less than $100$ and having exactly three positive divisors. Show that this product is a square.

2012 Mathcenter Contest + Longlist, 2 sl11
Tags: number theory , perfect square
Define the sequence of positive prime numbers. $p_1,p_2,p_3,...$. Let set $A$ be the infinite set of positive integers whose prime divisor does not exceed $p_n$. How many at least members must be selected from the set $A$ , such that we ensures that there are $2$ numbers whose products are perfect squares? [i](PP-nine)[/i]

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.

2017 JBMO Shortlist, NT3
Tags: number theory , perfect square , positive integer
Find all pairs of positive integers $(x,y)$ such that $2^x + 3^y$ is a perfect square.

2021 Saudi Arabia Training Tests, 38
Tags: perfect square , number theory
Prove that the set of all divisors of a positive integer which is not a perfect square can be divided into pairs so that in each pair, one number is divided by another.

2004 India IMO Training Camp, 3
Tags: number theory , perfect square
An integer $n$ is said to be [i]good[/i] if $|n|$ is not the square of an integer. Determine all integers $m$ with the following property: $m$ can be represented, in infinitely many ways, as a sum of three distinct good integers whose product is the square of an odd integer. [i]Proposed by Hojoo Lee, Korea[/i]

2017 Balkan MO Shortlist, N2
Tags: functional equation , functional equation in n , number theory , perfect square
Find all functions $f :Z_{>0} \to Z_{>0}$ such that the number $xf(x) + f ^2(y) + 2xf(y)$ is a perfect square for all positive integers $x,y$.

2024 Turkey Team Selection Test, 4
Tags: number theory , modular arithmetic , olympiad number theory , factorial , perfect square
Find all positive integer pairs $(a,b)$ such that, $$\frac{10^{a!} - 3^b +1}{2^a}$$ is a perfect square.

2021 Azerbaijan Senior NMO, 2
Tags: number theory , perfect square
Find all triples of natural numbers $(a, b, c)$ for which the number $$2^a + 2^b + 2^c + 3$$ is the square of an integer.

1997 All-Russian Olympiad Regional Round, 8.1
Tags: number theory , perfect square , combinatorics
Prove that the numbers from $1$ to $16$ can be written in a line, but cannot be written in a circle so that the sum of any two adjacent numbers is square of a natural number.

2018 Malaysia National Olympiad, B3
Tags: number theory , perfect square
Let $n$ be an integer greater than $1$, such that $3n + 1$ is a perfect square. Prove that $n + 1$ can be expressed as a sum of three perfect squares.

2011 Argentina National Olympiad, 4
Tags: number theory , perfect square
For each natural number $n$ we denote $a_n$ as the greatest perfect square less than or equal to $n$ and $b_n$ as the least perfect square greater than $n$. For example $a_9=3^2$, $b_9=4^2$ and $a_{20}=4^2$, $b_{20}=5^2$. Calculate: $$\frac{1}{a_1b_1}+\frac{1}{a_2b_2}+\frac{1}{a_3b_3}+\ldots +\frac{1}{a_{600}b_{600}}$$

2003 IMO Shortlist, 5
Tags: number theory , perfect square
An integer $n$ is said to be [i]good[/i] if $|n|$ is not the square of an integer. Determine all integers $m$ with the following property: $m$ can be represented, in infinitely many ways, as a sum of three distinct good integers whose product is the square of an odd integer. [i]Proposed by Hojoo Lee, Korea[/i]

2018 May Olympiad, 1
Tags: number theory , digit , perfect square
You have a $4$-digit whole number that is a perfect square. Another number is built adding $ 1$ to the unit's digit, subtracting $ 1$ from the ten's digit, adding $ 1$ to the hundred's digit and subtracting $ 1$ from the ones digit of one thousand. If the number you get is also a perfect square, find the original number. It's unique?

2019 Grand Duchy of Lithuania, 4
Tags: number theory , perfect square
Determine all pairs of prime numbers $(p, q)$ such that $p^2 + 5pq + 4q^2$ is a square of an integer.

2021 Durer Math Competition Finals, 2
Tags: number theory , perfect square
Find the number of integers $n$ between $1$ and $2021$ such that $2^n+2^{n+3}$ is a perfect square.

2022 Cyprus JBMO TST, 1
Tags: number theory , perfect square
Prove that for every natural number $k$, at least one of the integers \[ 2k-1, \quad 5k-1 \quad \text{and} \quad 13k-1\] is not a perfect square.

2009 IMO Shortlist, 7
Tags: number theory , perfect square , exponential
Let $a$ and $b$ be distinct integers greater than $1$. Prove that there exists a positive integer $n$ such that $(a^n-1)(b^n-1)$ is not a perfect square. [i]Proposed by Mongolia[/i]