Found problems: 171
2012 Bundeswettbewerb Mathematik, 2
Are there positive integers $a$ and $b$ such that both $a^2 + 4b$ and $b^2 + 4a$ are perfect squares?
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
\[\frac{a^p - a}{p}=b^2.\]
2013 Vietnam Team Selection Test, 2
a. Prove that there are infinitely many positive integers $t$ such that both $2012t+1$ and $2013t+1$ are perfect squares.
b. Suppose that $m,n$ are positive integers such that both $mn+1$ and $mn+n+1$ are perfect squares. Prove that $8(2m+1)$ divides $n$.
2022 Assara - South Russian Girl's MO, 1
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.
II Soros Olympiad 1995 - 96 (Russia), 10.5
Find all pairs of natural numbers $x$ and $y$ for which $x^2+3y$ and $y^2+3x$ are simultaneously squares of natural numbers.
2017 Hanoi Open Mathematics Competitions, 3
Suppose $n^2 + 4n + 25$ is a perfect square. How many such non-negative integers $n$'s are there?
(A): $1$ (B): $2$ (C): $4$ (D): $6$ (E): None of the above.
2021 German National Olympiad, 6
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.
2008 JBMO Shortlist, 5
Is it possible to arrange the numbers $1^1, 2^2,..., 2008^{2008}$ one after the other, in such a way that the obtained number is a perfect square? (Explain your answer.)
2009 District Olympiad, 3
Let $a$ and $b$ be non-negative integers. Prove that the number $a^2 + b^2$ is the difference of two perfect squares if and only if $ab$ is even.
2012 Mathcenter Contest + Longlist, 2 sl11
Define the sequence of positive prime numbers. $p_1,p_2,p_3,...$. Let set $A$ be the infinite set of positive integers whose prime divisor does not exceed $p_n$. How many at least members must be selected from the set $A$ , such that we ensures that there are $2$ numbers whose products are perfect squares?
[i](PP-nine)[/i]
2009 Abels Math Contest (Norwegian MO) Final, 1b
Show that the sum of three consecutive perfect cubes can always be written as the difference between two perfect squares.
2008 Brazil Team Selection Test, 4
Find all odd integers $n$ for which $\frac{2^{\phi (n)}-1}{n}$ is a perfect square.
2007 Alexandru Myller, 1
Solve $ x^3-y^3=2xy+7 $ in integers.
2024-IMOC, N1
Proof that for every primes $p$, $q$
\[p^{q^2-q+1}+q^{p^2-p+1}-p-q\]
is never a perfect square.
[i]Proposed by chengbilly[/i]
2016 Costa Rica - Final Round, A2
The initial number of inhabitants of a city of more than $150$ inhabitants is a perfect square. With an increase of $1000$ inhabitants it becomes a perfect square plus a unit. After from another increase of $1000$ inhabitants it is again a perfect square. Determine the quantity of inhabitants that are initially in the city.
1996 IMO Shortlist, 3
A finite sequence of integers $ a_0, a_1, \ldots, a_n$ is called quadratic if for each $ i$ in the set $ \{1,2 \ldots, n\}$ we have the equality $ |a_i \minus{} a_{i\minus{}1}| \equal{} i^2.$
a.) Prove that any two integers $ b$ and $ c,$ there exists a natural number $ n$ and a quadratic sequence with $ a_0 \equal{} b$ and $ a_n \equal{} c.$
b.) Find the smallest natural number $ n$ for which there exists a quadratic sequence with $ a_0 \equal{} 0$ and $ a_n \equal{} 1996.$
2016 Cono Sur Olympiad, 1
Let $\overline{abcd}$ be one of the 9999 numbers $0001, 0002, 0003, \ldots, 9998, 9999$. Let $\overline{abcd}$ be an [i]special[/i] number if $ab-cd$ and $ab+cd$ are perfect squares, $ab-cd$ divides $ab+cd$ and also $ab+cd$ divides $abcd$. For example 2016 is special. Find all the $\overline{abcd}$ special numbers.
[b]Note:[/b] If $\overline{abcd}=0206$, then $ab=02$ and $cd=06$.
2022 Grand Duchy of Lithuania, 4
Find all triples of natural numbers $(a, b, c)$ for which the number $$2^a + 2^b + 2^c + 3$$ is the square of an integer.
2024 OMpD, 4
Let \(a_0, a_1, a_2, \dots\) be an infinite sequence of positive integers with the following properties:
- \(a_0\) is a given positive integer;
- For each integer \(n \geq 1\), \(a_n\) is the smallest integer greater than \(a_{n-1}\) such that \(a_n + a_{n-1}\) is a perfect square.
For example, if \(a_0 = 3\), then \(a_1 = 6\), \(a_2 = 10\), \(a_3 = 15\), and so on.
(a) Let \(T\) be the set of numbers of the form \(a_k - a_l\), with \(k \geq l \geq 0\) integers.
Prove that, regardless of the value of \(a_0\), the number of positive integers not in \(T\) is finite.
(b) Calculate, as a function of \(a_0\), the number of positive integers that are not in \(T\).
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 $100$.
2024 Mexican University Math Olympiad, 1
Let \( x \), \( y \), \( p \) be positive integers that satisfy the equation \( x^4 = p + 9y^4 \), where \( p \) is a prime number. Show that \( \frac{p^2 - 1}{3} \) is a perfect square and a multiple of 16.