Olympiad problems

2022 Saudi Arabia JBMO TST, 4

Determine the smallest positive integer $a$ for which there exist a prime number $p$ and a positive integer $b \ge 2$ such that $$\frac{a^p -a}{p}=b^2.$$

2021 European Mathematical Cup, 3

Tags: factorial , perfect square , number theory

Let $\ell$ be a positive integer. We say that a positive integer $k$ is [i]nice [/i] if $k!+\ell$ is a square of an integer. Prove that for every positive integer $n \geqslant \ell$, the set $\{1, 2, \ldots,n^2\}$ contains at most $n^2-n +\ell$ nice integers. \\ \\ (Théo Lenoir)

2022 China Team Selection Test, 5

Tags: perfect square , number theory

Show that there exist constants $c$ and $\alpha > \frac{1}{2}$, such that for any positive integer $n$, there is a subset $A$ of $\{1,2,\ldots,n\}$ with cardinality $|A| \ge c \cdot n^\alpha$, and for any $x,y \in A$ with $x \neq y$, the difference $x-y$ is not a perfect square.

Russian TST 2020, P3

Tags: ceiling function , number theory , 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]

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 $3$.

2016 District Olympiad, 2

Tags: equation , algebra , perfect square , number theory , fractional part

If $ a,n $ are two natural numbers corelated by the equation $ \left\{ \sqrt{n+\sqrt n}\right\} =\left\{ \sqrt a\right\} , $ then $ 1+4a $ is a perfect square. Justify this statement. Here, $ \{\} $ is the usual fractionary part.

2008 Singapore Senior Math Olympiad, 2

Tags: number theory , perfect square

Determine all primes $p$ such that $5^p + 4 p^4$ is a perfect square, i.e., the square of an integer.

2001 All-Russian Olympiad Regional Round, 9.8

Tags: number theory , perfect square , digit

Sasha wrote a non-zero number on the board and added it to it on the right, one non-zero digit at a time, until he writes out a million digits. Prove that an exact square has been written on the board no more than $100$ times.

2020 Peru Cono Sur TST., P4

Tags: perfect square , number theory , phi function , euler s phi function

Find all odd integers $n$ for which $\frac{2^{\phi (n)}-1}{n}$ is a perfect square.

2006 Junior Balkan Team Selection Tests - Romania, 4

Tags: digit , number theory , perfect square

For a positive integer $n$ denote $r(n)$ the number having the digits of $n$ in reverse order- for example, $r(2006) = 6002$. Prove that for any positive integers a and b the numbers $4a^2 + r(b)$ and $4b^2 + r(a)$ can not be simultaneously squares.

1966 Poland - Second Round, 4

Tags: number theory , perfect square

Prove that if the natural numbers $ a $ and $ b $ satisfy the equation $ a^2+a = 3b^2 $, then the number $ a+1 $ is the square of an integer.

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

2020 Malaysia IMONST 1, 4

Tags: perfect square , number theory

This sequence lists the perfect squares in increasing order: \[0,1,4,9,16,\cdots ,a,10^8,b,\cdots\] Determine the value of $b-a$.

2019 Thailand TSTST, 1

Tags: number theory , perfect square

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

2018 Romania National Olympiad, 1

Tags: perfect square , sum of squares , number theory

Prove that there are infinitely many sets of four positive integers so that the sum of the squares of any three elements is a perfect square.

2012 Czech-Polish-Slovak Junior Match, 3

Tags: perfect square , number theory

Prove that if $n$ is a positive integer then $2 (n^2 + 1) - n$ is not a square of an integer.

2010 Contests, 1

Tags: perfect square , permutation

A [i]permutation[/i] of the set of positive integers $[n] = \{1, 2, . . . , n\}$ is a sequence $(a_1 , a_2 , \ldots, a_n ) $ such that each element of $[n]$ appears precisely one time as a term of the sequence. For example, $(3, 5, 1, 2, 4)$ is a permutation of $[5]$. Let $P (n)$ be the number of permutations of $[n]$ for which $ka_k$ is a perfect square for all $1 \leq k \leq n$. Find with proof the smallest $n$ such that $P (n)$ is a multiple of $2010$.

1986 IMO Longlists, 26

Tags: quadratic , modular arithmetic , number theory , system of equations , perfect square

Let $d$ be any positive integer not equal to $2, 5$ or $13$. Show that one can find distinct $a,b$ in the set $\{2,5,13,d\}$ such that $ab-1$ is not a perfect square.

1997 Abels Math Contest (Norwegian MO), 1

Tags: perfect square , sum , number theory

We call a positive integer $n$ [i]happy [/i] if there exist integers $a,b$ such that $a^2+b^2 = n$. If $t$ is happy, show that (a) $2t$ is [i]happy[/i], (b) $3t$ is not [i]happy[/i]

1966 Dutch Mathematical Olympiad, 3

Tags: perfect square , number theory

How many natural numbers are there whose square is a thirty-digit number which has the following curious property: If that thirty-digit number is divided from left to right into three groups of ten digits, then the numbers given by the middle group and the right group formed numbers are both four times the number formed by the left group?

2022 Cyprus JBMO TST, 1

Tags: perfect square , number theory

Find all integer values of $x$ for which the value of the expression \[x^2+6x+33\] is a perfect square.

1992 All Soviet Union Mathematical Olympiad, 570

Tags: sequence , integer sequence , perfect square , algebra , number theory

Define the sequence $a_1 = 1, a_2, a_3, ...$ by $$a_{n+1} = a_1^2 + a_2 ^2 + a_3^2 + ... + a_n^2 + n$$ Show that $1$ is the only square in the sequence.

2020 EGMO, 6

Tags: perfect square , sequence

Let $m > 1$ be an integer. A sequence $a_1, a_2, a_3, \ldots$ is defined by $a_1 = a_2 = 1$, $a_3 = 4$, and for all $n \ge 4$, $$a_n = m(a_{n - 1} + a_{n - 2}) - a_{n - 3}.$$ Determine all integers $m$ such that every term of the sequence is a square.

2009 Belarus Team Selection Test, 4

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

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

2009 Bosnia And Herzegovina - Regional Olympiad, 1

Tags: number theory , perfect cube , perfect square

Prove that for every positive integer $m$ there exists positive integer $n$ such that $m+n+1$ is perfect square and $mn+1$ is perfect cube of some positive integers