Olympiad problems

1984 Bundeswettbewerb Mathematik, 3

Let $a$ and $b$ be positive integers. Show that if $a \cdot b$ is even, then there are positive integers $c$ and $d$ with $a^2 + b^2 + c^2 = d^2$; if, on the other hand, $a\cdot b$ is odd, there are no such positive integers $c$ and $d$.

2004 Estonia National Olympiad, 3

Tags: number theory , Digits , Perfect Square

The teacher had written on the board a positive integer consisting of a number of $4$s followed by the same number of $8$s followed . During the break, Juku stepped up to the board and added to the number one more $4$ at the start and a $9$ at the end. Prove that the resulting number is an a square. of an integer.

2014 IFYM, Sozopol, 2

Tags: number theory , Perfect Square

Does there exist a natural number $n$, for which $n.2^{2^{2014}}-81-n$ is a perfect square?

2013 Kyiv Mathematical Festival, 5

Tags: number theory , Perfect power , Perfect Square

Do there exist positive integers $a \ne b$ such that $ a+b$ is a perfect square and $a^3 +b^3$ is a fourth power of an integer?

2002 Belarusian National Olympiad, 1

Tags: combinatorics , number theory , Perfect Square

Determine the largest possible number of groups one can compose from the integers $1,2,3,..., 19,20$, so that the product of the numbers in each group is a perfect square. (The group may contain exactly one number, in that case the product equals this number, each number must be in exactly one group.) (E. Barabanov, I. Voronovich)

2007 Greece JBMO TST, 2

Tags: number theory , Perfect Square , multiple

Let $n$ be a positive integer such that $n(n+3)$ is a perfect square of an integer, prove that $n$ is not a multiple of $3$.

2019 Portugal MO, 3

Tags: factorial , number theory , Perfect Square

The product $1\times 2\times 3\times ...\times n$ is written on the board. For what integers $n \ge 2$, we can add exclamation marks to some factors to convert them into factorials, in such a way that the final product can be a perfect square?

2014 Denmark MO - Mohr Contest, 4

Tags: number theory , Perfect Square

Determine all positive integers $n$ so that both $20n$ and $5n + 275$ are perfect squares. (A perfect square is a number which can be expressed as $k^2$, where $k$ is an integer.)

2007 Cuba MO, 2

Tags: number theory , Perfect Square

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.

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.

2023 Regional Olympiad of Mexico West, 4

Tags: number theory , Perfect Powers , Perfect power , Perfect Square , perfect cube

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.

2011 Saudi Arabia Pre-TST, 2.3

Tags: algebra , polynomial , Perfect Square

Let $f = aX^2 + bX+ c \in Z[X]$ be a polynomial such that for every positive integer $n$,$ f(n )$ is a perfect square. Prove that $f = g^2$ for some polynomial $g \in Z[X]$.

2015 Puerto Rico Team Selection Test, 6

Tags: number theory , Perfect Square

Find all positive integers $n$ such that $7^n + 147$ is a perfect square.

2009 IMAR Test, 4

Tags: number theory , Perfect Square , number theory with sequences , Sequence

Given any $n$ positive integers, and a sequence of $2^n$ integers (with terms among them), prove there exists a subsequence made of consecutive terms, such that the product of its terms is a perfect square. Also show that we cannot replace $2^n$ with any lower value (therefore $2^n$ is the threshold value for this property).

2019 Saudi Arabia Pre-TST + Training Tests, 3.3

Tags: Perfect Square , Product , recurrence relation , number theory , Sequence

Define sequence of positive integers $(a_n)$ as $a_1 = a$ and $a_{n+1} = a^2_n + 1$ for $n \ge 1$. Prove that there is no index $n$ for which $$\prod_{k=1}^{n} \left(a^2_k + a_k + 1\right)$$ is a perfect square.

2013 Denmark MO - Mohr Contest, 4

Tags: number theory , Perfect Square

The positive integer $a$ is greater than $10$, and all its digits are equal. Prove that $a$ is not a perfect square. (A perfect square is a number which can be expressed as $n^2$ , where $n$ is an integer.)

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)

1995 Tournament Of Towns, (475) 3

Tags: number theory , Perfect Square

The first digit of a $6$-digit number is $5$. Is it true that it is always possible to write $6$ more digits to the right of this number so that the resulting $12$-digit number is a perfect square? (A Tolpygo)

2014 Belarus Team Selection Test, 3

Tags: number theory , Perfect Square , decimal representation , IMO Shortlist , Hi

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.

2003 Singapore Senior Math Olympiad, 1

Tags: Perfect Square , divisible , divides , 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.

2020 Germany Team Selection Test, 3

Tags: ceiling function , number theory , IMO Shortlist , IMO Shortlist 2019 , Perfect Square

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]

1991 ITAMO, 2

Tags: Perfect Square , number theory

Prove that no number of the form $a^3+3a^2+a$, for a positive integer $a$, is a perfect square.

1996 Tournament Of Towns, (484) 2

Tags: number theory , Perfect Square

Does there exist an integer n such that all three numbers (a) $n - 96$, $n$ and $n + 96$ (b) $n - 1996$, $n$ and $n + 1996$ are positive prime numbers? (V Senderov)

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.

1992 Romania Team Selection Test, 2

Tags: linear algebra , matrix , Perfect Square , Sequence

For a positive integer $a$, define the sequence ($x_n$) by $x_1 = x_2 = 1$ and $x_{n+2 }= (a^4 +4a^2 +2)x_{n+1} -x_n -2a^2$ , for n $\ge 1$. Show that $x_n$ is a perfect square and that for $n > 2$ its square root equals the first entry in the matrix $\begin{pmatrix} a^2+1 & a \\ a & 1 \end{pmatrix}^{n-2}$