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

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

1993 Tournament Of Towns, (360) 3
Tags: number theory , perfect square , diophantine , diophantine equation
Positive integers $a$, $b$ and $c$ are positive integers with greatest common divisor equal to $1$ (i.e. they have no common divisors greater than $1$), and $$\frac{ab}{a-b}=c$$ Prove that $a -b$ is a perfect square. (SL Berlov)

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.

2019 India PRMO, 14
Tags: prime , perfect square , relation
Find the smallest positive integer $n \geq 10$ such that $n + 6$ is a prime and $9n + 7$ is a perfect square.

2021 Middle European Mathematical Olympiad, 8
Tags: perfect square , number theory , digit
Prove that there are infinitely many positive integers $n$ such that $n^2$ written in base $4$ contains only digits $1$ and $2$.

1998 Romania National Olympiad, 1
Tags: inequalities , algebra , perfect square , number theory
Let $n$ be a positive integer and $x_1,x_2,...,x_n$ be integer numbers such that $$x_1^2+x_2^2+...+x_n^2+ n^3 \le (2n - 1)(x_1+x_2+...+x_n ) + n^2$$ . Show that : a) $x_1,x_2,...,x_n$ are non-negative integers b) the number $x_1+x_2+...+x_n+n+1$ is not a perfect square.

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.

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]

1978 Czech and Slovak Olympiad III A, 6
Tags: number theory , algebra , sequence , perfect square
Show that the number \[p_n=\left(\frac{3+\sqrt5}{2}\right)^n+\left(\frac{3-\sqrt5}{2}\right)^n-2\] is a positive integer for any positive integer $n.$ Furthermore, show that the numbers $p_{2n-1}$ and $p_{2n}/5$ are perfect squares $($for any positive integer $n).$

2014 India IMO Training Camp, 2
Tags: number theory , perfect square , decimal representation
Determine whether there exists an infinite sequence of nonzero digits $a_1 , a_2 , a_3 , \cdots $ and a positive integer $N$ such that for every integer $k > N$, the number $\overline{a_k a_{k-1}\cdots a_1 }$ is a perfect square.

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

2003 Singapore Senior Math Olympiad, 1
Tags: perfect square , divisible , divide , number theory
It is given that n is a positive integer such that both numbers $2n + 1$ and $3n + 1$ are complete squares. Is it true that $n$ must be divisible by $40$ ? Justify your answer.

2002 Abels Math Contest (Norwegian MO), 1a
Tags: perfect square , number theory
Find all integers $k$ such that both $k + 1$ and $16k + 1$ are perfect squares.

2020 Durer Math Competition Finals, 3
Tags: number theory , least common multiple , perfect square , consecutive
Is it possible for the least common multiple of five consecutive positive integers to be a perfect square?

1964 All Russian Mathematical Olympiad, 054
Tags: number theory , perfect square
Find the smallest exact square with last digit not $0$, such that after deleting its last two digits we shall obtain another exact square.

1988 Greece National Olympiad, 4
Tags: perfect square , number theory
Prove that there are do not exist natural numbers $k, m$ such that numbers $k^2+2m$, $m^2+2k$ to be squares of integers.

1999 German National Olympiad, 6b
Tags: perfect square , number theory
Determine all pairs ($m,n$) of natural numbers for which $4^m + 5^n$ is a perfect square.

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?

2023 239 Open Mathematical Olympiad, 4
Tags: perfect square , number theory
We call a natural number [i]almost a square[/i] if it can be represented as a product of two numbers that differ by no more than one percent of the larger of them. Prove that there are infinitely many consecutive quadruples of almost squares.

2021 Brazil National Olympiad, 3
Tags: perfect square , power , number theory , floor function , irrational number
Find all positive integers \(k\) for which there is an irrational \(\alpha>1\) and a positive integer \(N\) such that \(\left\lfloor\alpha^{n}\right\rfloor\) is a perfect square minus \(k\) for every integer \(n\) with \(n>N\).

2011 Saudi Arabia Pre-TST, 1.3
Tags: perfect square , number theory
Find all positive integers $n$ such that $27^n- 2^n$ is a perfect square.