Olympiad problems

2022 Cyprus TST, 2

Let $n, m$ be positive integers such that \[n(4n+1)=m(5m+1)\] (a) Show that the difference $n-m$ is a perfect square of a positive integer. (b) Find a pair of positive integers $(n, m)$ which satisfies the above relation. Additional part (not asked in the TST): Find all such pairs $(n,m)$.

2010 IMO, 3

Tags: function , algebra , IMO Shortlist , IMO 2010 , Perfect Squares , number theory , IMO

Find all functions $g:\mathbb{N}\rightarrow\mathbb{N}$ such that \[\left(g(m)+n\right)\left(g(n)+m\right)\] is a perfect square for all $m,n\in\mathbb{N}.$ [i]Proposed by Gabriel Carroll, USA[/i]

2013 AIME Problems, 6

Tags: modular arithmetic , AMC , number theory , Perfect Squares , AIME

Find the least positive integer $N$ such that the set of $1000$ consecutive integers beginning with $1000 \cdot N$ contains no square of an integer.

2023 New Zealand MO, 1

Tags: number theory , Perfect Squares

For any positive integer $n$ let $n! = 1\times 2\times 3\times ... \times n$. Do there exist infinitely many triples $(p, q, r)$, of positive integers with $p > q > r > 1$ such that the product $p! \cdot q! \cdot r!$$ is a perfect square?

2013 Dutch IMO TST, 2

Tags: Perfect Squares , number theory , rational

Determine all integers $n$ for which $\frac{4n-2}{n+5}$ is the square of a rational number.

2022 Pan-African, 2

Tags: number theory , algebra , Perfect Squares

Find all $3$-tuples $(a, b, c)$ of positive integers, with $a \geq b \geq c$, such that $a^2 + 3b$, $b^2 + 3c$, and $c^2 + 3a$ are all squares.

2021 Federal Competition For Advanced Students, P2, 6

Tags: number theory , Perfect Square , Perfect Squares

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)

2010 IMO Shortlist, 5

Tags: function , algebra , IMO Shortlist , IMO 2010 , Perfect Squares , number theory , IMO

Find all functions $g:\mathbb{N}\rightarrow\mathbb{N}$ such that \[\left(g(m)+n\right)\left(g(n)+m\right)\] is a perfect square for all $m,n\in\mathbb{N}.$ [i]Proposed by Gabriel Carroll, USA[/i]

2024 JBMO TST - Turkey, 5

Tags: number theory , Divisibility , Perfect Squares

Find all positive integer values of $n$ such that the value of the $$\frac{2^{n!}-1}{2^n-1}$$ is a square of an integer.

2020 Colombia National Olympiad, 1

Tags: number theory , Digits , Perfect Squares , Perfect Square

A positive integer is called [i]sabroso [/i]if when it is added to the number obtained when its digits are interchanged from one side of its written form to the other, the result is a perfect square. For example, $143$ is sabroso, since $143 + 341 =484 = 22^2$. Find all two-digit sabroso numbers.

1990 IMO, 3

Tags: complex numbers , algebra , convex polygon , Perfect Squares , IMO , IMO 1990 , Harm Derksen

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.

Fractal Edition 2, P1

Tags: Perfect Squares

The positive integers $a$, $b$, $c$ are such that $\frac{a+b}{b+c}$ is the square of a rational number, and $ab+bc+ca$ is a prime number. Find all possible values of $\frac{a+b}{b+c}$.

1990 IMO Shortlist, 16

Tags: complex numbers , algebra , convex polygon , Perfect Squares , IMO , IMO 1990 , Harm Derksen

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.

Fractal Edition 1, P3

Tags: Perfect Squares

Can the number $ \overline{abc} + \overline{bca} + \overline{cab} $ be a perfect square?

1991 Bundeswettbewerb Mathematik, 1

Tags: number theory , Perfect Squares , Diophantine equation , diophantine

Determine all solutions of the equation $4^x + 4^y + 4^z = u^2$ for integers $x,y,z$ and $u$.

2022 Abelkonkurransen Finale, 1a

Tags: number theory , Perfect Squares

Determine all positive integers $n$ such that $2022 + 3^n$ is a perfect square.

1994 IMO Shortlist, 1

Tags: number theory , Extremal combinatorics , Perfect Squares , Subset , IMO Shortlist

$ M$ is a subset of $ \{1, 2, 3, \ldots, 15\}$ such that the product of any three distinct elements of $ M$ is not a square. Determine the maximum number of elements in $ M.$

2020 Romanian Master of Mathematics Shortlist, N2

Tags: number theory , Perfect Squares , RMM , RMM 2020 , RMM Shortlist

For a positive integer $n$, let $\varphi(n)$ and $d(n)$ denote the value of the Euler phi function at $n$ and the number of positive divisors of $n$, respectively. Prove that there are infinitely many positive integers $n$ such that $\varphi(n)$ and $d(n)$ are both perfect squares. [i]Finland, Olli Järviniemi[/i]

2021 Argentina National Olympiad Level 2, 5

Tags: number theory , Perfect Squares , contests

Determine all positive integers $n$ such that $$n\cdot 2^{n-1}+1$$ is a perfect square.

2009 Belarus Team Selection Test, 1

Tags: Perfect Squares , Perfect Square , number theory

Prove that there exist many natural numbers n so that both roots of the quadratic equation $x^2+(2-3n^2)x+(n^2-1)^2=0$ are perfect squares. S. Kuzmich

1996 Dutch Mathematical Olympiad, 2

Tags: Perfect Squares , 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.

2015 Thailand TSTST, 1

Tags: number theory , square , Perfect Squares , Sum of Squares

Prove that there exist infinitely many integers $n$ such that $n, n + 1, n + 2$ are each the sum of two squares of integers.

2020 Malaysia IMONST 1, 4

Tags: Perfect Squares , number theory

This sequence lists the perfect squares in increasing order: \[0,1,4,9,16,\cdots ,a,10^8,b,\cdots\] Determine the value of $b-a$.

2004 India IMO Training Camp, 3

Tags: number theory , Perfect Squares , IMO Shortlist

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 Thailand Mathematical Olympiad, 5

Tags: number theory , Perfect Squares , Sum of Squares , consecutive

Does there exist $2017$ consecutive positive integers, none of which could be written as $a^2 + b^2$ for some integers $a, b$? Justify your answer.