Olympiad problems

2017 CHMMC (Fall), 5

Find the number of primes $p$ such that $p! + 25p$ is a perfect square.

1984 IMO Longlists, 25

Tags: number theory , product , perfect square , equation

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

2020 Dutch Mathematical Olympiad, 4

Tags: number theory , perfect square , prime

Determine all pairs of integers $(x, y)$ such that $2xy$ is a perfect square and $x^2 + y^2$ is a prime number.

2008 Flanders Math Olympiad, 2

Tags: number theory , perfect square

Let $a, b$ and $c$ be integers such that $a+b+c = 0$. Prove that $\frac12(a^4 +b^4 +c^4)$ is a perfect square.

2021 Regional Olympiad of Mexico West, 2

Tags: number theory , perfect square

Prove that in every $16$-digit number there is a chain of one or more consecutive digits such that the product of those digits is a perfect square. For example, if the original number is $7862328578632785$ we can take the digits $6$, $2$ and $3$ whose product is $6^2$ (note that these appear consecutively in the number).

2024 Tuymaada Olympiad, 3

Tags: perfect square , sequence , algebra

All perfect squares, and all perfect squares multiplied by two, are written in a row in increasing order. let $f(n)$ be the $n$-th number in this sequence. (For instance, $f(1)=1,f(2)=2,f(3)=4,f(4)=8$.) Is there an integer $n$ such that all the numbers \[f(n),f(2n),f(3n),\dots,f(10n^2)\] are perfect squares?

2018 Malaysia National Olympiad, B3

Tags: perfect square , number theory

Let $n$ be an integer greater than $1$, such that $3n + 1$ is a perfect square. Prove that $n + 1$ can be expressed as a sum of three perfect squares.

1999 Bundeswettbewerb Mathematik, 4

Tags: perfect cube , perfect square , number theory

A natural number is called [i]bright [/i] if it is the sum of a perfect square and a perfect cube. Prove that if $r$ and $s$ are any two positive integers, then (a) there exist infinitely many positive integers $n$ such that both $r+n$ and $s+n$ are [i]bright[/i], (b) there exist infinitely many positive integers $m$ such that both rm and sm are [i]bright[/i].

1997 Tournament Of Towns, (562) 3

Tags: number theory , integer , perfect square

All expressions of the form $$\pm \sqrt1 \pm \sqrt2 \pm ... \pm \sqrt{100}$$ (with every possible combination of signs) are multiplied together. Prove that the result is: (a) an integer; (b) the square of an integer. (A Kanel)

1988 IMO, 3

Tags: number theory , divisibility , diophantine equation , perfect square , vieta jumping

Let $ a$ and $ b$ be two positive integers such that $ a \cdot b \plus{} 1$ divides $ a^{2} \plus{} b^{2}$. Show that $ \frac {a^{2} \plus{} b^{2}}{a \cdot b \plus{} 1}$ is a perfect square.

2017 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: perfect square , arranging , number theory , combinatorics

Prove that numbers $1,2,...,16$ can be divided in sequence such that sum of any two neighboring numbers is perfect square

2013 Nordic, 1

Tags: number theory , perfect square , integer sequence , floor function

Let ${(a_n)_{n\ge1}} $ be a sequence with ${a_1 = 1} $ and ${a_{n+1} = \lfloor a_n +\sqrt{a_n}+\frac{1}{2}\rfloor }$ for all ${n \ge 1}$, where ${\lfloor x \rfloor}$ denotes the greatest integer less than or equal to ${x}$. Find all ${n \le 2013}$ such that ${a_n}$ is a perfect square

1994 All-Russian Olympiad Regional Round, 9.7

Tags: number theory , perfect square

Find all prime numbers $p,q,r,s$ such that their sum is a prime number and $p^2+qs$ and $p^2 +qr$ are squares of integers.

2011 District Olympiad, 3

Tags: number theory , perfect square

A positive integer $N$ has the digits $1, 2, 3, 4, 5, 6$ and $7$, so that each digit $i$, $i \in \{1, 2, 3, 4, 5, 6, 7\}$ occurs $4i$ times in the decimal representation of $N$. Prove that $N$ is not a perfect square.

2012 Argentina National Olympiad, 2

Tags: number theory , perfect square

Determine all natural numbers $n$ for which there are $2n$ distinct positive integers $x_1,…,x_n,y_1,…,y_n$ such that the product $$(11x^2_1+12y^2_1)(11x^2_2+12y^2_2)…(11x^2_n+12y^2_n)$$ is a perfect square.

2012 Estonia Team Selection Test, 1

Tags: perfect square , perfect cube , number theory

Prove that for any positive integer $k$ there exist $k$ pairwise distinct integers for which the sum of their squares equals the sum of their cubes.

1999 Switzerland Team Selection Test, 10

Tags: product , consecutive , perfect square , number theory

Prove that the product of five consecutive positive integers cannot be a perfect square.

2024 Turkey Team Selection Test, 4

Tags: modular arithmetic , olympiad number theory , factorial , perfect square , number theory

Find all positive integer pairs $(a,b)$ such that, $$\frac{10^{a!} - 3^b +1}{2^a}$$ is a perfect square.

2023 Regional Olympiad of Mexico West, 4

Tags: perfect cube , perfect square , perfect power , number theory

Prove that you can pick $15$ distinct positive integers between $1$ and $2023$, such that each one of them and the sum between some of them is never a perfect square, nor a perfect cube or any other greater perfect power.

2020 Greece Team Selection Test, 4

Tags: perfect square , ceiling function , 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]

1989 Swedish Mathematical Competition, 1

Tags: digit , perfect square , number theory

Let $n$ be a positive integer. Prove that the numbers $n^2(n^2 + 2)^2$ and $n^4(n^2 + 2)^2$ are written in base $n^2 +1$ with the same digits but in opposite order.

1988 IMO Longlists, 76

Tags: binomial theorem , perfect square , algebra , number theory

A positive integer is called a [b]double number[/b] if its decimal representation consists of a block of digits, not commencing with 0, followed immediately by an identical block. So, for instance, 360360 is a double number, but 36036 is not. Show that there are infinitely many double numbers which are perfect squares.

2017 JBMO Shortlist, NT3

Tags: positive integer , perfect square , number theory

Find all pairs of positive integers $(x,y)$ such that $2^x + 3^y$ is a perfect square.

2021 Malaysia IMONST 1, 15

Tags: perfect square , number theory

Find the sum of all integers $n$ with this property: both $n$ and $n + 2021$ are perfect squares.

2020 Junior Balkan Team Selection Tests - Moldova, 4

Tags: perfect square , number theory

A natural number $n$ is called "$k$-squared" if it can be written as a sum of $k$ perfect squares not equal to 0. a) Prove that 2020 is "$2$-squared" , "$3$-squared" and "$4$-squared". b) Determine all natural numbers not equal to 0 ($a, b, c, d ,e$) $a<b<c<d<e$ that verify the following conditions simultaneously : 1) $e-2$ , $e$ , $e+4$ are all prime numbers. 2) $a^2+ b^2 + c^2 + d^2 + e^2$ = 2020.