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

2022 Assara - South Russian Girl's MO, 1
Tags: perfect square , number theory
Given three natural numbers $a$, $b$ and $c$. It turned out that they are coprime together. And their least common multiple and their product are perfect squares. Prove that $a$, $b$ and $c$ are perfect squares.

2013 Denmark MO - Mohr Contest, 4
Tags: perfect square , number theory
The positive integer $a$ is greater than $10$, and all its digits are equal. Prove that $a$ is not a perfect square. (A perfect square is a number which can be expressed as $n^2$ , where $n$ is an integer.)

1997 Singapore Senior Math Olympiad, 3
Tags: digit , perfect square , number theory
Find the smallest positive integer $x$ such that $x^2$ ends with the four digits $9009$.

2021 Argentina National Olympiad Level 2, 5
Tags: perfect square , number theory
Determine all positive integers $n$ such that $$n\cdot 2^{n-1}+1$$ is a perfect square.

2021 Austrian MO National Competition, 6
Tags: perfect square , number theory
Let $p$ be an odd prime number and $M$ a set derived from $\frac{p^2 + 1}{2}$ square numbers. Investigate whether $p$ elements can be selected from this set whose arithmetic mean is an integer. (Walther Janous)

2001 Singapore Team Selection Test, 1
Tags: number theory , perfect square
Let $a, b, c, d$ be four positive integers such that each of them is a difference of two squares of positive integers. Prove that $abcd$ is also a difference of two squares of positive integers.

2014 Denmark MO - Mohr Contest, 4
Tags: perfect square , number theory
Determine all positive integers $n$ so that both $20n$ and $5n + 275$ are perfect squares. (A perfect square is a number which can be expressed as $k^2$, where $k$ is an integer.)

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.

2007 JBMO Shortlist, 3
Tags: number theory , perfect square
Let $n > 1$ be a positive integer and $p$ a prime number such that $n | (p - 1) $and $p | (n^6 - 1)$. Prove that at least one of the numbers $p- n$ and $p + n$ is a perfect square.

2015 Peru MO (ONEM), 3
Tags: floor function , perfect square , number theory
Let $a_1, a_2, . . . , a_n$ be positive integers, with $n \ge 2$, such that $$ \lfloor \sqrt{a_1 \cdot a_2\cdot\cdot\cdot a_n} \rfloor = \lfloor \sqrt{a_1} \rfloor \cdot \lfloor \sqrt{a_2} \rfloor \cdot\cdot\cdot \lfloor \sqrt{a_n} \rfloor.$$ Prove that at least $n - 1$ of these numbers are perfect squares. Clarification: Given a real number $x$, $\lfloor x\rfloor$ denotes the largest integer that is less than or equal to $x$. For example $\lfloor \sqrt2\rfloor$ and $\lfloor 3\rfloor =3$.

1997 Bundeswettbewerb Mathematik, 4
Tags: number theory , perfect square , divisible
Prove that if $n$ is a natural number such that both $3n+1$ and $4n+1$ are squares, then $n$ is divisible by $56$.

2009 IMO Shortlist, 7
Tags: perfect square , exponential , number theory
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]

1996 Dutch Mathematical Olympiad, 2
Tags: perfect square , number theory
Investigate whether for two positive integers $m$ and $n$ the numbers $m^2 + n$ and $n^2 + m$ can be both squares of integers.

2016 Poland - Second Round, 4
Tags: number theory , set , perfect square
Let $k$ be a positive integer. Show that exists positive integer $n$, such that sets $A = \{ 1^2, 2^2, 3^2, ...\}$ and $B = \{1^2 + n, 2^2 + n, 3^2 + n, ... \}$ have exactly $k$ common elements.

2010 All-Russian Olympiad Regional Round, 11.4
Tags: perfect square , number theory
We call a triple of natural numbers $(a, b, c)$ [i]square [/i] if they form an arithmetic progression (in exactly this order), the number $b$ is coprime to each of the numbers $a$ and $c$, and the number $abc$ is a perfect square. Prove that for any given a square triple, there is another square triple that has at least one common number with it.

2017 Peru IMO TST, 4
Tags: perfect square , number theory , factorial
The product $1\times 2\times 3\times ...\times n$ is written on the board. For what integers $n \ge 2$, we can add exclamation marks to some factors to convert them into factorials, in such a way that the final product can be a perfect square?

2018 Saudi Arabia IMO TST, 2
Tags: perfect square , number theory , combinatorics
Let $n$ be an even positive integer. We fill in a number on each cell of a rectangle table of $n$ columns and multiple rows as following: i. Each row is assigned to some positive integer $a$ and its cells are filled by $0$ or $a$ (in any order); ii. The sum of all numbers in each row is $n$. Note that we cannot add any more row to the table such that the conditions (i) and (ii) still hold. Prove that if the number of $0$’s on the table is odd then the maximum odd number on the table is a perfect square.

2010 Puerto Rico Team Selection Test, 5
Tags: number theory , perfect square
Find all prime numbers $p$ and $q$ such that $2p^2q + 45pq^2$ is a perfect square.

2010 Thailand Mathematical Olympiad, 1
Tags: number theory , perfect square
Show that, for every positive integer $x$, there is a positive integer $y\in \{2, 5, 13\}$ such that $xy - 1$ is not a perfect square.

2021 Denmark MO - Mohr Contest, 3
Tags: perfect square , number theory
Georg investigates which integers are expressible in the form $$\pm 1^2 \pm 2^2 \pm 3^2 \pm \dots \pm n^2.$$ For example, the number $3$ can be expressed as $ -1^2 + 2^2$, and the number $-13$ can be expressed as $+1^2 + 2^2 + 3^2 - 4^2 + 5^2 - 6^2$. Are all integers expressible in this form?

1996 IMO Shortlist, 2
Tags: modular arithmetic , perfect square , number theory , quadratic reciprocity
The positive integers $ a$ and $ b$ are such that the numbers $ 15a \plus{} 16b$ and $ 16a \minus{} 15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?

1990 IMO Shortlist, 16
Tags: perfect square , algebra , complex number , convex polygon
Prove that there exists a convex 1990-gon with the following two properties : [b]a.)[/b] All angles are equal. [b]b.)[/b] The lengths of the 1990 sides are the numbers $ 1^2$, $ 2^2$, $ 3^2$, $ \cdots$, $ 1990^2$ in some order.

2014 Junior Balkan Team Selection Tests - Moldova, 5
Tags: number theory , floor function , perfect square
Show that for any natural number $n$, the number $A = [\frac{n + 3}{4}] + [ \frac{n + 5}{4} ] + [\frac{n}{2} ] +n^2 + 3n + 3$ is a perfect square. ($[x]$ denotes the integer part of the real number x.)

2020 Germany Team Selection Test, 3
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]

1992 Chile National Olympiad, 1
Tags: perfect square , number theory
Determine all naturals $n$ such that $2^n + 5$ is a perfect square.