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

2018 Greece National Olympiad, 1
Tags: sequence , perfect square
Let $(x_n), n\in\mathbb{N}$ be a sequence such that $x_{n+1}=3x_n^3+x_n, \forall n\in\mathbb{N}$ and $x_1=\frac{a}{b}$ where $a,b$ are positive integers such that $3\not|b$. If $x_m$ is a square of a rational number for some positive integer $m$, prove that $x_1$ is also a square of a rational number.

2003 IMO Shortlist, 5
Tags: perfect square , number theory
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]

2008 Singapore Junior Math Olympiad, 5
Tags: number theory , perfect square
Determine all primes $p$ such that $5^p + 4 p^4$ is a perfect square, i.e., the square of an integer.

2021 German National Olympiad, 6
Tags: arithmetic progression , perfect square , number theory
Determine whether there are infinitely many triples $(u,v,w)$ of positive integers such that $u,v,w$ form an arithmetic progression and the numbers $uv+1, vw+1$ and $wu+1$ are all perfect squares.

OIFMAT III 2013, 9
Tags: perfect square , number theory
Let $ a, b \in Z $, prove that if the expression $ a \cdot 2013^n + b $ is a perfect square for all natural $n$, then $ a $ is zero.

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: $$\frac{1}{a_1b_1}+\frac{1}{a_2b_2}+\frac{1}{a_3b_3}+\ldots +\frac{1}{a_{600}b_{600}}$$

2005 Singapore Senior Math Olympiad, 1
Tags: perfect square , digit , difference , number theory
The digits of a $3$-digit number are interchanged so that none of the digits retain their original position. The difference of the two numbers is a $2$-digit number and is a perfect square. Find the difference.

2020 Czech and Slovak Olympiad III A, 4
Tags: number theory , perfect square
Positive integers $a, b$ satisfy equality $b^2 = a^2 + ab + b$. Prove that $b$ is a square of a positive integer. (Patrik Bak)

1999 Austrian-Polish Competition, 5
Tags: recurrence relation , sequence , perfect square , number theory
A sequence of integers $(a_n)$ satisfies $a_{n+1} = a_n^3 + 1999$ for $n = 1,2,....$ Prove that there exists at most one $n$ for which $a_n$ is a perfect square.

India EGMO 2021 TST, 6
Tags: number theory , perfect square
Let $n>2$ be a positive integer and $b=2^{2^n}$. Let $a$ be an odd positive integer such that $a\le b \le 2a$. Show that $a^2+b^2-ab$ is not a square.

2010 Dutch IMO TST, 3
Tags: number theory , perfect square
(a) Let $a$ and $b$ be positive integers such that $M(a, b) = a - \frac1b +b(b + \frac3a)$ is an integer. Prove that $M(a,b)$ is a square. (b) Find nonzero integers $a$ and $b$ such that $M(a,b)$ is a positive integer, but not a square.

2023 China Northern MO, 6
Tags: number theory , perfect square
A positive integer $m$ is called a [i]beautiful [/i] integer if that there exists a positive integer $n$ such that $m = n^2+ n + 1$. Prove that there are infinitely many [i]beautiful [/i] integers with square factors, and the square factors of different beautiful integers are relative prime.

2020 Federal Competition For Advanced Students, P1, 4
Tags: perfect square , number theory
Determine all positive integers $N$ such that $$2^N-2N$$ is a perfect square. (Walther Janous)

2001 China Team Selection Test, 1
Tags: perfect square , number theory
For which integer \( h \), are there infinitely many positive integers \( n \) such that \( \lfloor \sqrt{h^2 + 1} \cdot n \rfloor \) is a perfect square? (Here \( \lfloor x \rfloor \) denotes the integer part of the real number \( x \)?

2022 Latvia Baltic Way TST, P13
Tags: number theory , perfect square
Call a pair of integers $a$ and $b$ square makers , if $ab+1$ is a perfect square. Determine for which $n$ is it possible to divide the set $\{1,2, \dots , 2n\}$ into $n$ pairs of square makers.

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

1940 Moscow Mathematical Olympiad, 062-
Tags: number theory , perfect square
Find a four-digit number that is perfect square and such that the first two digits are the same and the last two as well.

2007 Dutch Mathematical Olympiad, 4
Tags: perfect square , number theory , exponential
Determine the number of integers $a$ satisfying $1 \le a \le 100$ such that $a^a$ is a perfect square. (And prove that your answer is correct.)

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

2013 Dutch IMO TST, 2
Tags: perfect square , number theory , rational
Determine all integers $n$ for which $\frac{4n-2}{n+5}$ is the square of a rational number.

2022 Durer Math Competition Finals, 4
Tags: perfect square , number theory , divisor
Show that the divisors of a number $n \ge 2$ can only be divided into two groups in which the product of the numbers is the same if the product of the divisors of $n$ is a square number.

2013 Czech-Polish-Slovak Match, 1
Tags: perfect square , trinomial , number theory , integer polynomial
Let $a$ and $b$ be integers, where $b$ is not a perfect square. Prove that $x^2 + ax + b$ may be the square of an integer only for finite number of integer values of $x$. (Martin Panák)

2009 IMAR Test, 4
Tags: number theory , perfect square , number theory with sequences , sequence
Given any $n$ positive integers, and a sequence of $2^n$ integers (with terms among them), prove there exists a subsequence made of consecutive terms, such that the product of its terms is a perfect square. Also show that we cannot replace $2^n$ with any lower value (therefore $2^n$ is the threshold value for this property).

1984 IMO Longlists, 25
Tags: number theory , equation , product , perfect square
Prove that the product of five consecutive positive integers cannot be the square of an integer.

1974 Putnam, A3
Tags: number theory , perfect square , prime number
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) $4x^2 +4xy+ 5y^2.$