Found problems: 521
1998 Belarus Team Selection Test, 2
a) Given that integers $a$ and $b$ satisfy the equality $$a^2 - (b^2 - 4b + 1) a - (b^4 - 2b^3) = 0 \,\,\, (*)$$, prove that $b^2 + a$ is a square of an integer.
b) Do there exist an infinitely many of pairs $(a,b)$ satisfying (*)?
2008 Singapore Senior Math Olympiad, 2
Determine all primes $p$ such that $5^p + 4 p^4$ is a perfect square, i.e., the square of an integer.
2020 Junior Balkan Team Selection Tests - Moldova, 10
Find all pairs of prime numbers $(p, q)$ for which the numbers $p+q$ and $p+4q$ are simultaneously perfect squares.
1992 Tournament Of Towns, (348) 6
Consider the sequence $a(n)$ defined by the following conditions: $$a(1) = 1\,\,\,\, a(n + 1) = a(n) + [\sqrt{a(n)}] \,\,\, , \,\,\,\, n = 1,2,3,...$$
Prove that the sequence contains an infinite number of perfect squares. (Note: $[x]$ means the integer part of $x$, that is the greatest integer not greater than $x$.)
(A Andjans)
2015 Silk Road, 2
Let $\left\{ {{a}_{n}} \right\}_{n \geq 1}$ and $\left\{ {{b}_{n}} \right\}_{n \geq 1}$ be two infinite arithmetic progressions, each of which the first term and the difference are mutually prime natural numbers. It is known that for any natural $n$, at least one of the numbers $\left( a_n^2+a_{n+1}^2 \right)\left( b_n^2+b_{n+1}^2 \right) $ or $\left( a_n^2+b_n^2 \right) \left( a_{n+1}^2+b_{n+1}^2 \right)$ is an perfect square. Prove that ${{a}_{n}}={{b}_{n}}$, for any natural $n$ .
2016 Estonia Team Selection Test, 11
Find all positive integers $n$ such that $(n^2 + 11n - 4) \cdot n! + 33 \cdot 13^n + 4$ is a perfect square
2012 Flanders Math Olympiad, 2
Let $n$ be a natural number. Call $a$ the smallest natural number you need to subtract from $n$ to get a perfect square. Call $b$ the smallest natural number that you must add to $n$ to get a perfect square. Prove that $n - ab$ 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.
2009 Thailand Mathematical Olympiad, 1
Let $S \subset Z^+$ be a set of positive integers with the following property: for any $a, b \in S$, if $a \ne b$ then $a + b$ is a perfect square. Given that $2009 \in S$ and $2087 \in S$, what is the maximum number of elements in $S$?
2000 Singapore Team Selection Test, 2
Find all prime numbers $p$ such that $5^p + 12^p$ is a perfect square
2021 IMO Shortlist, N2
Let $n \geqslant 100$ be an integer. Ivan writes the numbers $n, n+1, \ldots, 2 n$ each on different cards. He then shuffles these $n+1$ cards, and divides them into two piles. Prove that at least one of the piles contains two cards such that the sum of their numbers is a perfect square.
2021 Belarusian National Olympiad, 10.5
Prove that for any positive integer $n$ there exist infinitely many triples $(a,b,c)$ of pairwise distinct positive integers such that $ab+n,bc+n,ac+n$ are all perfect squares
2022 Korea National Olympiad, 3
Suppose that the sequence $\{a_n\}$ of positive integers satisfies the following conditions:
[list]
[*]For an integer $i \geq 2022$, define $a_i$ as the smallest positive integer $x$ such that $x+\sum_{k=i-2021}^{i-1}a_k$ is a perfect square.
[*]There exists infinitely many positive integers $n$ such that $a_n=4\times 2022-3$.
[/list]
Prove that there exists a positive integer $N$ such that $\sum_{k=n}^{n+2021}a_k$ is constant for every integer $n \geq N$.
And determine the value of $\sum_{k=N}^{N+2021}a_k$.
2020 Kosovo National Mathematical Olympiad, 2
Find all positive integers $x$, $y$ such that $2^x+5^y+2$ is a perfect square.
1979 IMO Longlists, 69
Let $N$ be the number of integral solutions of the equation
\[x^2 - y^2 = z^3 - t^3\]
satisfying the condition $0 \leq x, y, z, t \leq 10^6$, and let $M$ be the number of integral solutions of the equation
\[x^2 - y^2 = z^3 - t^3 + 1\]
satisfying the condition $0 \leq x, y, z, t \leq 10^6$. Prove that $N >M.$
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
2010 Austria Beginners' Competition, 1
Prove that $2010$ cannot be represented as the difference between two square numbers.
(B. Schmidt, Graz University of Technology)
1994 Argentina National Olympiad, 2
For what positive integer values of $x$ is $x^4 + 6x^3 + 11x^2 + 3x + 31$ a perfect square?
2019 Canadian Mathematical Olympiad Qualification, 4
Let $n$ be a positive integer. For a positive integer $m$, we partition the set $\{1, 2, 3,...,m\}$ into $n$ subsets, so that the product of two different elements in the same subset is never a perfect square. In terms of $n$, fi nd the largest positive integer $m$ for which such a partition exists.
2004 Germany Team Selection Test, 3
Let $ b$ be an integer greater than $ 5$. For each positive integer $ n$, consider the number \[ x_n = \underbrace{11\cdots1}_{n \minus{} 1}\underbrace{22\cdots2}_{n}5, \] written in base $ b$.
Prove that the following condition holds if and only if $ b \equal{} 10$: [i]there exists a positive integer $ M$ such that for any integer $ n$ greater than $ M$, the number $ x_n$ is a perfect square.[/i]
[i]Proposed by Laurentiu Panaitopol, Romania[/i]
1979 IMO Shortlist, 21
Let $N$ be the number of integral solutions of the equation
\[x^2 - y^2 = z^3 - t^3\]
satisfying the condition $0 \leq x, y, z, t \leq 10^6$, and let $M$ be the number of integral solutions of the equation
\[x^2 - y^2 = z^3 - t^3 + 1\]
satisfying the condition $0 \leq x, y, z, t \leq 10^6$. Prove that $N >M.$
1986 IMO Longlists, 26
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.
2019 AMC 10, 11
How many positive integer divisors of $201^9$ are perfect squares or perfect cubes (or both)?
$\textbf{(A) } 32 \qquad\textbf{(B) } 36 \qquad\textbf{(C) } 37 \qquad\textbf{(D) } 39 \qquad\textbf{(E) } 41$
2010 Contests, 3
Find all functions $g:\mathbb{N}\rightarrow\mathbb{N}$ such that \[\left(g(m)+n\right)\left(g(n)+m\right)\] is a perfect square for all $m,n\in\mathbb{N}.$
[i]Proposed by Gabriel Carroll, USA[/i]
2014 IFYM, Sozopol, 2
Does there exist a natural number $n$, for which $n.2^{2^{2014}}-81-n$ is a perfect square?