Olympiad problems

2018 Bosnia And Herzegovina - Regional Olympiad, 2

Find all positive integers $n$ such that number $n^4-4n^3+22n^2-36n+18$ is perfect square of positive integer

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 All-Russian Olympiad Regional Round, 9.2

Tags: algebra , perfect square

Rational numbers $a$ and $b$ satisfy the equality $$a^3b+ab^3+2a^2b^2+2a + 2b + 1 = 0. $$ Prove that the number $1-ab$ is the square of the rational numbers.

1984 IMO Shortlist, 10

Tags: number theory , equation , product , perfect square

Prove that the product of five consecutive positive integers cannot be the square of an integer.

2010 Puerto Rico Team Selection Test, 5

Tags: perfect square , number theory

Find all prime numbers $p$ and $q$ such that $2p^2q + 45pq^2$ 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)

1966 Swedish Mathematical Competition, 3

Tags: sum of squares , number theory , perfect square

Show that an integer $= 7 \mod 8$ cannot be sum of three squares.

1963 Swedish Mathematical Competition., 1

Tags: number theory , perfect square

How many positive integers have square less than $10^7$?

2019 AMC 10, 11

Tags: perfect square , perfect cube

How many positive integer divisors of $201^9$ are perfect squares or perfect cubes (or both)? $\textbf{(A) } 32 \qquad\textbf{(B) } 36 \qquad\textbf{(C) } 37 \qquad\textbf{(D) } 39 \qquad\textbf{(E) } 41$

2005 Estonia National Olympiad, 5

Tags: perfect square , number theory

Does there exist an integer $n > 1$ such that $2^{2^n-1} -7$ is not a perfect square?

1953 Kurschak Competition, 2

Tags: number theory , perfect square , divide

$n$ and $d$ are positive integers such that $d$ divides $2n^2$. Prove that $n^2 + d$ cannot be a square.

2020 Regional Olympiad of Mexico West, 3

Tags: perfect square , number theory

Prove that for every natural number $ n>2 $ there exists an integer $ k $ that can be written as the sum of $ i $ positive perfect squares, for every $ i $ between $ 2 $ and $ n $.

2020 Olympic Revenge, 4

Tags: perfect square , combinatorics , number theory , additive combinatorics

Let $n$ be a positive integer and $A$ a set of integers such that the set $\{x = a + b\ |\ a, b \in A\}$ contains $\{1^2, 2^2, \dots, n^2\}$. Prove that there is a positive integer $N$ such that if $n \ge N$, then $|A| > n^{0.666}$.

1938 Eotvos Mathematical Competition, 1

Tags: number theory , perfect square

Prove that an integer $n$ can be expressed as the sum of two squares if and only if $2n$ can be expressed as the sum of two squares.

2010 Contests, 3

Tags: perfect square , function , algebra , number theory

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]

1995 Rioplatense Mathematical Olympiad, Level 3, 1

Tags: number theory , perfect square

Let $n$ and $p$ be two integers with $p$ positive prime, such that $pn + 1$ is a perfect square. Show that $n + 1$ is the sum of $p$ perfect squares, not necessarily distinct.

2025 All-Russian Olympiad, 9.3

Tags: perfect square , number theory

Find all natural numbers $n$ for which there exists an even natural number $a$ such that the number \[ (a - 1)(a^2 - 1)\cdots(a^n - 1) \] is a perfect square.

2018 Bosnia And Herzegovina - Regional Olympiad, 3

Tags: number theory , prime number , perfect square

Let $p$ and $q$ be prime numbers such that $p^2+pq+q^2$ is perfect square. Prove that $p^2-pq+q^2$ is prime

1995 Tournament Of Towns, (444) 4

Tags: number theory , perfect square

Prove that the number $\overline{40...0}9$ (with at least one zero) is not a perfect square. (VA Senderov)

2015 Greece JBMO TST, 3

Tags: number theory , perfect square , positive integer

Prove that there is not a positive integer $n$ such that numbers $(n+1)2^n, (n+3)2^{n+2}$ are both perfect squares.

2021 Argentina National Olympiad, 2

Tags: perfect square , positive integer , number theory , prime number

Let $m$ be a positive integer for which there exists a positive integer $n$ such that the multiplication $mn$ is a perfect square and $m- n$ is prime. Find all $m$ for $1000\leq m \leq 2021.$

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.

1951 Moscow Mathematical Olympiad, 199

Tags: number theory , perfect square

Prove that the sum $1^3 + 2^3 +...+ n^3$ is a perfect square for all $n$.

2019 Tournament Of Towns, 5

Tags: perfect square , number theory

Let us say that the pair $(m, n)$ of two positive different integers m and n is [i]nice [/i] if $mn$ and $(m + 1)(n + 1)$ are perfect squares. Prove that for each positive integer m there exists at least one $n > m$ such that the pair $(m, n)$ is nice. (Yury Markelov)

2025 Bulgarian Spring Mathematical Competition, 10.3

Tags: table , perfect square , algebra

In the cell $(i,j)$ of a table $n\times n$ is written the number $(i-1)n + j$. Determine all positive integers $n$ such that there are exactly $2025$ rows not containing a perfect square.