Olympiad problems

2021 European Mathematical Cup, 3

Tags: factorial , Perfect Square , number theory , emc , European Mathematical Cup

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)

2004 Switzerland Team Selection Test, 2

Tags: Perfect Square , number theory

Find the largest natural number $n$ for which $4^{995} +4^{1500} +4^n$ is a square.

2021 Brazil Undergrad MO, Problem 3

Tags: Brazilian Undergrad MO 2021 , real analysis , powers , irrational number , approximation , Perfect Square

Find all positive integers $k$ for which there is an irrational $\alpha>1$ and a positive integer $N$ such that $\left\lfloor\alpha^{n}\right\rfloor$ is of the form $m^2-k$ com $m \in \mathbb{Z}$ for every integer $n>N$.

2010 Dutch IMO TST, 3

Tags: number theory , Perfect Square

Let $n\ge 2$ be a positive integer and $p $ a prime such that $n|p-1$ and $p | n^3-1$. Show $ 4p-3$ is a square.

2014 Hanoi Open Mathematics Competitions, 4

Tags: number theory , Perfect Square , minimum

Find the smallest positive integer $n$ such that the number $2^n + 2^8 + 2^{11}$ is a perfect square. (A): $8$, (B): $9$, (C): $11$, (D): $12$, (E) None of the above.

2015 Turkey MO (2nd round), 1

Tags: number theory , Turkey , Divisibility , Perfect Square

$m$ and $n$ are positive integers. If the number \[ k=\dfrac{(m+n)^2}{4m(m-n)^2+4}\] is an integer, prove that $k$ is a perfect square.

2012 Cuba MO, 8

Tags: number theory , Perfect Square

If the natural numbers $a, b, c, d$ verify the relationships: $$(a^2 + b^2)(c^2 + d^2) = (ab + cd)^2$$ $$(a^2 + d^2)(b^2 + c^2) = (ad + bc)^2$$ and the $gcd(a, b, c, d) = 1$, prove that $a + b + c + d$ is a perfect square.

1988 IMO, 3

Tags: number theory , Divisibility , Diophantine equation , Perfect Square , Vieta Jumping , IMO , IMO 1988

Let $ a$ and $ b$ be two positive integers such that $ a \cdot b \plus{} 1$ divides $ a^{2} \plus{} b^{2}$. Show that $ \frac {a^{2} \plus{} b^{2}}{a \cdot b \plus{} 1}$ is a perfect square.

2022 Assara - South Russian Girl's MO, 1

Tags: number theory , Perfect Squares , Perfect Square

Given three natural numbers $a$, $b$ and $c$. It turned out that they are coprime together. And their least common multiple and their product are perfect squares. Prove that $a$, $b$ and $c$ are perfect squares.

2021 Denmark MO - Mohr Contest, 3

Tags: number theory , Perfect Square

Georg investigates which integers are expressible in the form $$\pm 1^2 \pm 2^2 \pm 3^2 \pm \dots \pm n^2.$$ For example, the number $3$ can be expressed as $ -1^2 + 2^2$, and the number $-13$ can be expressed as $+1^2 + 2^2 + 3^2 - 4^2 + 5^2 - 6^2$. Are all integers expressible in this form?

II Soros Olympiad 1995 - 96 (Russia), 10.5

Tags: number theory , Perfect Squares , Perfect Square

Find all pairs of natural numbers $x$ and $y$ for which $x^2+3y$ and $y^2+3x$ are simultaneously squares of natural numbers.

2001 Denmark MO - Mohr Contest, 4

Tags: number theory , Perfect Square

Show that any number of the form $$4444 ...44 88...8$$ where there are twice as many $4$s as $8$s is a square number.

2021 German National Olympiad, 6

Tags: number theory , number theory proposed , Arithmetic Progression , Perfect Squares , Perfect Square

Determine whether there are infinitely many triples $(u,v,w)$ of positive integers such that $u,v,w$ form an arithmetic progression and the numbers $uv+1, vw+1$ and $wu+1$ are all perfect squares.

2020 China National Olympiad, 5

Tags: number theory , Perfect Square

Given any positive integer $c$, denote $p(c)$ as the largest prime factor of $c$. A sequence $\{a_n\}$ of positive integers satisfies $a_1>1$ and $a_{n+1}=a_n+p(a_n)$ for all $n\ge 1$. Prove that there must exist at least one perfect square in sequence $\{a_n\}$.

2021 Regional Olympiad of Mexico West, 2

Tags: number theory , Perfect Square

Prove that in every $16$-digit number there is a chain of one or more consecutive digits such that the product of those digits is a perfect square. For example, if the original number is $7862328578632785$ we can take the digits $6$, $2$ and $3$ whose product is $6^2$ (note that these appear consecutively in the number).

2008 IMAC Arhimede, 1

Tags: number theory , prime , Perfect Square

Find all prime numbers $ p $ for which $ 1 + p\cdot 2^{p} $ is a perfect square.

2016 Singapore Junior Math Olympiad, 1

Tags: Perfect Square , number theory

Find all integers$ n$ such that $n^2 + 24n + 35$ is a square.

1992 Chile National Olympiad, 1

Tags: Perfect Square , number theory

Determine all naturals $n$ such that $2^n + 5$ is a perfect square.

2011 May Olympiad, 2

Tags: Perfect Square , perfect cubes , number theory , Digits

Using only once each of the digits $1, 2, 3, 4, 5, 6, 7$ and $ 8$, write the square and the cube of a positive integer. Determine what that number can be.

2014 Junior Balkan Team Selection Tests - Moldova, 8

Tags: combinatorics , Perfect Square , number theory

The teacher wrote a non-zero natural number on the board. The teacher explained students that they can delete the number written on the board and can write a number instead naturally new, whenever they want, applying one of the following each time rules: 1) Instead of the current number $n$ write $3n + 13$ 2) Instead of the current number $n$ write the number $\sqrt{n}$, if $n$ is a perfect square . a) If the number $256$ was originally written on the board, is it possible that after a finite number of steps to get the number $55$ on the board? b) If the number $55$ was originally written on the board, is it possible that after a number finished the steps to get the number $256$ on the board?

1974 Swedish Mathematical Competition, 6

Tags: number theory , Perfect Square , Sum of Squares

For which $n$ can we find positive integers $a_1,a_2,\dots,a_n$ such that \[ a_1^2+a_2^2+\cdots+a_n^2 \] is a 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]

2006 Belarusian National Olympiad, 8

Tags: number theory , Perfect Square

a) Do there exist positive integers $a$ and $b$ such that for any positive,integer $n$ the number $a \cdot 2^n+ b\cdot 5^n$ is a perfect square ? b) Do there exist positive integers $a, b$ and $c$, such that for any positive integer $n$ the number $a\cdot 2^n+ b\cdot 5^n + c$ is a perfect square? (M . Blotski)

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.

2009 Abels Math Contest (Norwegian MO) Final, 1b

Tags: number theory , perfect cube , Perfect Square , Perfect Squares

Show that the sum of three consecutive perfect cubes can always be written as the difference between two perfect squares.