Olympiad problems

2011 Korea Junior Math Olympiad, 3

Let $x, y$ be positive integers such that $gcd(x, y) = 1$ and $x + 3y^2$ is a perfect square. Prove that $x^2 + 9y^4$ can't be a perfect square.

2015 India PRMO, 6

Tags: perfect square , number theory , digit , sum of digits

$6.$ How many two digit positive integers $N$ have the property that the sum of $N$ and the number obtained by reversing the order of the digits of $N$ is a perfect square $?$

1990 Bundeswettbewerb Mathematik, 2

Tags: sequence , perfect square , recurrence relation , algebra

The sequence $a_0,a_1,a_2,...$ is defined by $a_0 = 0, a_1 = a_2 = 1$ and $a_{n+2} +a_{n-1} = 2(a_{n+1} +a_n)$ for all $n \in N$. Show that all $a_n$ are perfect squares .

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)

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.

2019 Thailand TSTST, 1

Tags: number theory , perfect square

Find all primes $p$ such that $(p-3)^p+p^2$ is a perfect square.

1977 All Soviet Union Mathematical Olympiad, 244

Tags: number theory , perfect square

Let us call "fine" the $2n$-digit number if it is exact square itself and the two numbers represented by its first $n$ digits (first digit may not be zero) and last $n$ digits (first digit may be zero, but it may not be zero itself) are exact squares also. a) Find all two- and four-digit fine numbers. b) Is there any six-digit fine number? c) Prove that there exists $20$-digit fine number. d) Prove that there exist at least ten $100$-digit fine numbers. e) Prove that there exists $30$-digit fine number.

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)

1988 Tournament Of Towns, (168) 1

Tags: number theory , perfect square

We are given that $a, b$ and $c$ are whole numbers (i.e. positive integers) . Prove that if $a = b + c$ then $a^4 + b^4 + c^4$ is double the square of a whole number. (Folklore)

1965 Dutch Mathematical Olympiad, 2

Tags: number theory , perfect square , divide , divisible

Prove that $S_1 = (n + 1)^2 + (n + 2)^2 +...+ (n + 5)^2$ is divisible by $5$ for every $n$. Prove that for no $n$: $\sum_{\ell=1}^5 (n+\ell)^2$ is a perfect square. Let $S_2=(n + 6)^2 + (n + 7)^2 + ... + (n + 10)^2$. Prove that $S_1 \cdot S_2$ is divisible by $150$.

2011 Denmark MO - Mohr Contest, 1

Tags: combinatorics , number theory , perfect square

Georg writes the numbers from $1$ to $15$ on different pieces of paper. He attempts to sort these pieces of paper into two stacks so that none of the stacks contains two numbers whose sum is a square number.Prove that this is impossible. (The square numbers are the numbers $0 = 0^2$, $1 = 1^2$, $4 = 2^2$, $9 = 3^2$ etc.)

2021 South Africa National Olympiad, 3

Tags: number theory , prime , sum of squares , power of 2 , perfect square

Determine the smallest integer $k > 1$ such that there exist $k$ distinct primes whose squares sum to a power of $2$.

1995 Austrian-Polish Competition, 7

Tags: number theory , squarefree , perfect square , diophantine , diophantine equation

Consider the equation $3y^4 + 4cy^3 + 2xy + 48 = 0$, where $c$ is an integer parameter. Determine all values of $c$ for which the number of integral solutions $(x,y)$ satisfying the conditions (i) and (ii) is maximal: (i) $|x|$ is a square of an integer; (ii) $y$ is a squarefree number.

2010 Dutch BxMO TST, 5

Tags: number theory , quadratic , permutation , 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 $\{0,1,..., n\}$.

2013 Saudi Arabia BMO TST, 5

Tags: sum of squares , digit , perfect square , number theory

We call a positive integer [i]good[/i ] if it doesn’t have a zero digit and the sum of the squares of its digits is a perfect square. For example, $122$ and $34$ are good and $304$ and $12$ are not not good. Prove that there exists a $n$-digit good number for every positive integer $n$.

2005 Estonia National Olympiad, 5

Tags: number theory , perfect square

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

2019 Tournament Of Towns, 5

Tags: number theory , perfect square

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)

2016 Estonia Team Selection Test, 11

Tags: number theory , perfect square

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

2014 Finnish National High School Mathematics, 5

Tags: number theory , combinatorics , minimum , sum , perfect square

Determine the smallest number $n \in Z_+$, which can be written as $n = \Sigma_{a\in A}a^2$, where $A$ is a finite set of positive integers and $\Sigma_{a\in A}a= 2014$. In other words: what is the smallest positive number which can be written as a sum of squares of different positive integers summing to $2014$?

2009 Belarus Team Selection Test, 4

Tags: number theory , perfect square , diophantine , diophantine equation

Let $x,y,z$ be integer numbers satisfying the equality $yx^2+(y^2-z^2)x+y(y-z)^2=0$ a) Prove that number $xy$ is a perfect square. b) Prove that there are infinitely many triples $(x,y,z)$ satisfying the equality. I.Voronovich

2000 IMO Shortlist, 6

Tags: number theory , additive number theory , sum of squares , perfect square

Show that the set of positive integers that cannot be represented as a sum of distinct perfect squares is finite.