Olympiad problems

2019 Durer Math Competition Finals, 3

Determine all triples $(p, q, r)$ of prime numbers for which $p^q + p^r$ is a perfect square.

2018 India PRMO, 15

Tags: perfect square , minimum , number theory

Let $a$ and $b$ be natural numbers such that $2a-b$, $a-2b$ and $a+b$ are all distinct squares. What is the smallest possible value of $b$ ?

2020 Germany Team Selection Test, 3

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]

2021 Saudi Arabia JBMO TST, 4

Tags: perfect square , number theory

Let us call a set of positive integers nice if the number of its elements equals to the average of its numbers. Call a positive integer $n$ an [i]amazing[/i] number if the set $\{1, 2 , . . . , n\}$ can be partitioned into nice subsets. a) Prove that every perfect square is amazing. b) Show that there are infinitely many positive integers which are not amazing.

2018 Stars of Mathematics, 2

Tags: perfect square , divisible , number theory

Show that, if $m$ and $n$ are non-zero integers of like parity, and $n^2 -1$ is divisible by $m^2 - n^2 + 1$, then $m^2 - n^2 + 1$ is the square of an integer. Amer. Math. Monthly

2013 Vietnam Team Selection Test, 2

Tags: infinitely many solutions , perfect square , divisibility , function , number theory

a. Prove that there are infinitely many positive integers $t$ such that both $2012t+1$ and $2013t+1$ are perfect squares. b. Suppose that $m,n$ are positive integers such that both $mn+1$ and $mn+n+1$ are perfect squares. Prove that $8(2m+1)$ divides $n$.

2021 Saudi Arabia Training Tests, 38

Tags: perfect square , number theory

Prove that the set of all divisors of a positive integer which is not a perfect square can be divided into pairs so that in each pair, one number is divided by another.

2012 Mathcenter Contest + Longlist, 2 sl11

Tags: perfect square , number theory

Define the sequence of positive prime numbers. $p_1,p_2,p_3,...$. Let set $A$ be the infinite set of positive integers whose prime divisor does not exceed $p_n$. How many at least members must be selected from the set $A$ , such that we ensures that there are $2$ numbers whose products are perfect squares? [i](PP-nine)[/i]

2018 Malaysia National Olympiad, A5

Tags: perfect square , difference of squares , number theory

Determine the value of $(101 \times 99)$ - $(102 \times 98)$ + $(103 \times 97)$ − $(104 \times 96)$ + ... ... + $(149 \times 51)$ − $(150 \times 50)$.

Fractal Edition 1, P3

Tags: perfect square

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

2003 IMO Shortlist, 4

Tags: decimal representation , perfect square , number theory , modular arithmetic

Let $ b$ be an integer greater than $ 5$. For each positive integer $ n$, consider the number \[ x_n = \underbrace{11\cdots1}_{n \minus{} 1}\underbrace{22\cdots2}_{n}5, \] written in base $ b$. Prove that the following condition holds if and only if $ b \equal{} 10$: [i]there exists a positive integer $ M$ such that for any integer $ n$ greater than $ M$, the number $ x_n$ is a perfect square.[/i] [i]Proposed by Laurentiu Panaitopol, Romania[/i]

1997 IMO Shortlist, 15

Tags: perfect cube , perfect square , arithmetic sequence , number theory , modular arithmetic

An infinite arithmetic progression whose terms are positive integers contains the square of an integer and the cube of an integer. Show that it contains the sixth power of an integer.

2007 Cuba MO, 2

Tags: perfect square , number theory

Find three different positive integers whose sum is minimum than meet the condition that the sum of each pair of them is a perfect square.

1965 Polish MO Finals, 4

Tags: perfect square , number theory

Prove that if the integers $ a $ and $ b $ satisfy the equation $$ 2a^2 + a = 3b^2 + b,$$ then the numbers $ a - b $ and $ 2a + 2b + 1 $ are squares of integers.

2016 Bosnia and Herzegovina Junior BMO TST, 1

Tags: perfect square , number theory

Prove that it is not possible that numbers $(n+1)\cdot 2^n$ and $(n+3)\cdot 2^{n+2}$ are perfect squares, where $n$ is positive integer.

1953 Kurschak Competition, 2

Tags: perfect square , divide , number theory

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

2011 QEDMO 9th, 1

Tags: number theory , perfect square

Find all integers $n$ for which both $4n + 1$ and $9n + 1$ are perfect squares.

2000 Moldova Team Selection Test, 9

Tags: perfect square , sequence , number theory with sequences , integer

The sequence $x_{n}$ is defined by: $x_{0}=1, x_{1}=0, x_{2}=1,x_{3}=1, x_{n+3}=\frac{(n^2+n+1)(n+1)}{n}x_{n+2}+(n^2+n+1)x_{n+1}-\frac{n+1}{n}x_{n} (n=1,2,3..)$ Prove that all members of the sequence are perfect squares.

2016 Estonia Team Selection Test, 11

Tags: perfect square , number theory

Find all positive integers $n$ such that $(n^2 + 11n - 4) \cdot n! + 33 \cdot 13^n + 4$ is a perfect square

2008 Postal Coaching, 4

Tags: perfect square , perfect cube , number theory

Show that for each natural number $n$, there exist $n$ distinct natural numbers whose sum is a square and whose product is a cube.

2016 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: divisibility , fbh , perfect square , number theory

Let $a$ and $b$ be two positive integers such that $2ab$ divides $a^2+b^2-a$. Prove that $a$ is perfect square

2015 Caucasus Mathematical Olympiad, 5

Tags: number theory , perfect square , sides of a triangle

Are there natural $a, b >1000$ , such that for any $c$ that is a perfect square, the three numbers $a, b$ and $c$ are not the lengths of the sides of a triangle?

2008 JBMO Shortlist, 5

Tags: perfect square , number theory

Is it possible to arrange the numbers $1^1, 2^2,..., 2008^{2008}$ one after the other, in such a way that the obtained number is a perfect square? (Explain your answer.)

2005 iTest, 27

Tags: perfect square , digit , number theory

Find the sum of all non-zero digits that can repeat at the end of a perfect square. (For example, if $811$ were a perfect square, $1$ would be one of these non-zero digits.)