Found problems: 521
2006 Chile National Olympiad, 4
Let $n$ be a $6$-digit number, perfect square and perfect cube, if $n -6$ is neither even nor multiple of $3$. Find $n$ .
1988 IMO Shortlist, 9
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.
2020 Regional Olympiad of Mexico Northeast, 4
Let \(n > 1\) be an integer and \(p\) be a prime. Prove that if \(n|p-1\) and \(p|n^3-1\), then \(4p-3\) is a perfect square.
1952 Moscow Mathematical Olympiad, 224
a) Prove that if the square of a number begins with $0.\underbrace{\hbox{9...9}}_{\hbox{100}}$, then the number itself begins with $0.\underbrace{\hbox{9...9}}_{\hbox{100}}$,.
b) Calculate $\sqrt{0.9...9}$ ($60$ nines) to $60$ decimal places
2015 Turkey MO (2nd round), 1
$m$ 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.
2020 Junior Balkan Team Selection Tests - Moldova, 4
A natural number $n$ is called "$k$-squared" if it can be written as a sum of $k$ perfect squares not equal to 0.
a) Prove that 2020 is "$2$-squared" , "$3$-squared" and "$4$-squared".
b) Determine all natural numbers not equal to 0 ($a, b, c, d ,e$) $a<b<c<d<e$ that verify the following conditions simultaneously :
1) $e-2$ , $e$ , $e+4$ are all prime numbers.
2) $a^2+ b^2 + c^2 + d^2 + e^2$ = 2020.
2018 Rioplatense Mathematical Olympiad, Level 3, 1
Determine if there are $2018$ different positive integers such that the sum of their squares is a perfect cube and the sum of their cubes is a perfect square.
2021 Federal Competition For Advanced Students, P2, 6
Let $p$ be an odd prime number and $M$ a set derived from $\frac{p^2 + 1}{2}$ square numbers.
Investigate whether $p$ elements can be selected from this set whose arithmetic mean is an integer.
(Walther Janous)
1992 IMO, 3
For each positive integer $\,n,\;S(n)\,$ is defined to be the greatest integer such that, for every positive integer $\,k\leq S(n),\;n^{2}\,$ can be written as the sum of $\,k\,$ positive squares.
[b]a.)[/b] Prove that $\,S(n)\leq n^{2}-14\,$ for each $\,n\geq 4$.
[b]b.)[/b] Find an integer $\,n\,$ such that $\,S(n)=n^{2}-14$.
[b]c.)[/b] Prove that there are infintely many integers $\,n\,$ such that $S(n)=n^{2}-14.$
2006 Estonia National Olympiad, 2
Let $a, b$ and $c$ be positive integers such that $ab + 1, bc + 1$ and $ca + 1$ are all integer squares.
a) Give an example of such numbers $a, b$ and $c$.
b) Prove that at least one of the numbers $a, b$ and $c$ is divisible by $4$
2019 IMO Shortlist, N8
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]
2001 Romania Team Selection Test, 4
Show that the set of positive integers that cannot be represented as a sum of distinct perfect squares is finite.
2005 Austrian-Polish Competition, 4
Determine the smallest natural number $a\geq 2$ for which there exists a prime number $p$ and a natural number $b\geq 2$ such that
\[\frac{a^p - a}{p}=b^2.\]
2021 Brazil Undergrad MO, Problem 3
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 $n>N$.
2001 Cuba MO, 3
Let $n$ be a positive integer.
a) Prove that the number $(2n + 1)^3 - (2n - 1)^3$ is the sum of three perfect squares.
b) Prove that the number $(2n+1)^3-2$ is the sum of $3n-1$ perfect squares greater than $1$.
2013 Kyiv Mathematical Festival, 5
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?
2018 Junior Balkan Team Selection Tests - Romania, 1
Determine the prime numbers $p$ for which the number $a = 7^p - p - 16$ is a perfect square.
Lucian Petrescu
2021 Bangladeshi National Mathematical Olympiad, 7
For a positive integer $n$, let $s(n)$ and $c(n)$ be the number of divisors of $n$ that are perfect squares and perfect cubes respectively. A positive integer $n$ is called fair if $s(n)=c(n)>1$. Find the number of fair integers less than $100$.
2021 South Africa National Olympiad, 3
Determine the smallest integer $k > 1$ such that there exist $k$ distinct primes whose squares sum to a power of $2$.
2021 Malaysia IMONST 1, 5
How many integers $n$ (with $1 \le n \le 2021$) have the property that $8n + 1$ is a perfect square?
2015 Caucasus Mathematical Olympiad, 5
Are there natural $a, b >1000$ , such that for any $c$ that is a perfect square, the three numbers $a, b$ and $c$ are not the lengths of the sides of a triangle?
2023 Stars of Mathematics, 1
Determine all pairs $(p,q)$ of prime numbers for which $p^2+5pq+4q^2$ is a perfect square.
1987 Mexico National Olympiad, 4
Calculate the product of all positive integers less than $100$ and having exactly three positive divisors. Show that this product is a square.
2003 Junior Tuymaada Olympiad, 2
Find all natural $ x $ for which $ 3x+1 $ and $ 6x-2 $ are perfect squares, and the number $ 6x^2-1 $ is prime.
2023 New Zealand MO, 1
For any positive integer $n$ let $n! = 1\times 2\times 3\times ... \times n$. Do there exist infinitely many triples $(p, q, r)$, of positive integers with $p > q > r > 1$ such that the product $p! \cdot q! \cdot r!$$ is a perfect square?