Olympiad problems

2020 Germany Team Selection Test, 3

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]

2010 Saudi Arabia BMO TST, 1

Tags: Perfect Square , number theory

Find all integers $n$ for which $9n + 16$ and $16n + 9$ are both perfect squares.

1974 Putnam, A3

Tags: Putnam , number theory , Perfect Square , prime numbers

A well-known theorem asserts that a prime $p > 2$ can be written as the sum of two perfect squares ($p = m^2 +n^2$ , with $m$ and $n$ integers) if and only if $p \equiv 1$ (mod $4$). Assuming this result, find which primes $p > 2$ can be written in each of the following forms, using integers $x$ and $y$: a) $x^2 +16y^2, $ b) $4x^2 +4xy+ 5y^2.$

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

1989 Tournament Of Towns, (235) 3

Tags: number theory , Perfect Square , combinatorics

Do there exist $1000 000$ distinct positive integers such that the sum of any collection of these numbers is never an exact square?

2018 Polish Junior MO Finals, 1

Tags: number theory , Perfect Square , powers , Poland

Positive odd integers $a, b$ are such that $a^bb^a$ is a perfect square. Show that $ab$ is a perfect square.

2016 India PRMO, 16

Tags: algebra , geometric mean , arithmetic mean , Perfect Square , number theory

For positive real numbers $x$ and $y$, define their special mean to be average of their arithmetic and geometric means. Find the total number of pairs of integers $(x, y)$, with $x \le y$, from the set of numbers $\{1,2,...,2016\}$, such that the special mean of $x$ and $y$ 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

1999 All-Russian Olympiad Regional Round, 10.5

Tags: number theory , Perfect Square

Are there $10$ different integers such that all the sums made up of $9$ of them are perfect squares?

2014 India IMO Training Camp, 2

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.

2006 Chile National Olympiad, 4

Tags: number theory , Perfect Square , perfect cube

Let $n$ be a $6$-digit number, perfect square and perfect cube, if $n -6$ is neither even nor multiple of $3$. Find $n$ .

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.

2018 Ukraine Team Selection Test, 11

Tags: Perfect Square , combinatorics

$2n$ students take part in a math competition. First, each of the students sends its task to the members of the jury, after which each of the students receives from the jury one of proposed tasks (all received tasks are different). Let's call the competition [i]honest[/i], if there are $n$ students who were given the tasks suggested by the remaining $n$ participants. Prove that the number of task distributions in which the competition is honest is a square of natural numbers.

2012 Hanoi Open Mathematics Competitions, 7

Tags: number theory , Perfect Square

Prove that the number $a =\overline{{1...1}{5...5}6}$ is a perfect square (where $1$s are $2012$ in total and $5$s are $2011$ in total)

2010 Dutch BxMO TST, 5

Tags: permutation , Quadratic , number theory , Perfect Square

For any non-negative integer $n$, we say that a permutation $(a_0,a_1,...,a_n)$ of $\{0,1,..., n\} $ is quadratic if $k + a_k$ is a square for $k = 0, 1,...,n$. Show that for any non-negative integer $n$, there exists a quadratic permutation of $\{0,1,..., n\}$.

2002 Abels Math Contest (Norwegian MO), 1a

Tags: Perfect Square , number theory

Find all integers $k$ such that both $k + 1$ and $16k + 1$ are perfect squares.

1945 Moscow Mathematical Olympiad, 093

Tags: 2-digit , number theory , Perfect Square

Find all two-digit numbers $\overline {ab}$ such that $\overline {ab} + \overline {ba}$ 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.

1966 All Russian Mathematical Olympiad, 074

Tags: number theory , Perfect Square

Can both $(x^2+y)$ and $(y^2+x)$ be exact squares for natural $x$ and $y$?

2020 Durer Math Competition Finals, 3

Tags: number theory , least common multiple , Perfect Square , consecutive

Is it possible for the least common multiple of five consecutive positive integers to be a perfect square?

2015 Thailand Mathematical Olympiad, 8

Tags: Perfect Square , odd , number theory

Let $m$ and $n$ be positive integers such that $m - n$ is odd. Show that $(m + 3n)(5m + 7n)$ is not a perfect square.

2018 Junior Balkan Team Selection Tests - Romania, 1

Tags: number theory , Perfect Square

Determine the prime numbers $p$ for which the number $a = 7^p - p - 16$ is a perfect square. Lucian Petrescu

2005 Abels Math Contest (Norwegian MO), 1a

Tags: number theory , Sum , Perfect Square

A positive integer $m$ is called triangular if $m = 1 + 2 + ... + n$, for an integer $n$. Show that a positive integer $m$ is triangular if and only if $8m + 1$ is the square of an integer.

1988 IMO Longlists, 94

Tags: number theory , prime numbers , Perfect Square , IMO Longlist

Let $n+1, n \geq 1$ positive integers be formed by taking the product of $n$ given prime numbers (a prime number can appear several times or also not appear at all in a product formed in this way.) Prove that among these $n+1$ one can find some numbers whose product is a perfect square.

2022 Bulgaria National Olympiad, 3

Tags: number theory , coprime numbers , Perfect Square

Let $x>y>2022$ be positive integers such that $xy+x+y$ is a perfect square. Is it possible for every positive integer $z$ from the interval $[x+3y+1,3x+y+1]$ the numbers $x+y+z$ and $x^2+xy+y^2$ not to be coprime?