Found problems: 521
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)
Find all primes $p$ such that $\frac{2^{p-1}-1}{p}$ is a 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.
Determine all pairs of integers $(x, y)$ such that $2xy$ is a perfect square and $x^2 + y^2$ is a prime number.
Find all natural numbers \(n\) for which there exists an even natural number \(a\) such that the number
\[
(a - 1)(a^2 - 1)\cdots(a^n - 1)
\]
is a perfect square.
Prove that there are infinitely many positive integers $n$ such that $n^2$ written in base $4$ contains only digits $1$ and $2$.
Let $n+1, n \geq 1$ positive integers be formed by taking the product of $n$ given prime numbers (a prime number can appear several times or also not appear at all in a product formed in this way.) Prove that among these $n+1$ one can find some numbers whose product is a perfect square.
a) Let $f(x,y) = x^3 + (3y^2+1)x^2 + (3y^4 - y^2 + 4 y - 1)x + (y^6-y^4 + 2y^3)$. Prove that if for some positive integers $a, b$ the number $f(a, b)$ is a cube of an integer then $f(a, b)$ is also a square of an integer.
b) Are there infinitely many pairs of positive integers $(a, b)$ for which $f(a, b)$ is a square but not a cube ?
Prove that the number $\underbrace{111\ldots 11}_{1997}\underbrace{22\ldots 22}_{1998}5$ (which has 1997 of 1-s and 1998 of 2-s) is a perfect square.
The teacher had written on the board a positive integer consisting of a number of $4$s followed by the same number of $8$s followed . During the break, Juku stepped up to the board and added to the number one more $4$ at the start and a $9$ at the end. Prove that the resulting number is an a square. of an integer.
This sequence lists the perfect squares in increasing order: \[0,1,4,9,16,\cdots ,a,10^8,b,\cdots\]
Determine the value of $b-a$.
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.
A perfect square ends with the same two digits. How many possible values
of this digit are there?
Let $x, y$ be positive integers such that $gcd(x, y) = 1$ and $x + 3y^2$ is a perfect square. Prove that $x^2 + 9y^4$ can't be a perfect square.
We are given that $a, b$ and $c$ are whole numbers (i.e. positive integers) . Prove that if $a = b + c$ then $a^4 + b^4 + c^4$ is double the square of a whole number.
(Folklore)
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.$
The positive integers $ a$ and $ b$ are such that the numbers $ 15a \plus{} 16b$ and $ 16a \minus{} 15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?
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]
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.
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.
Find all pairs of positive integers $(x,y)$ such that $2^x + 3^y$ is a perfect square.
Find all prime numbers $p$ for which $p^3- p + 1$ is a perfect square .
A triple integer $(a, b, c)$ is called [i]brilliant [/i] when it satisfies:
(i) $a> b> c$ are prime numbers
(ii) $a = b + 2c$
(iii) $a + b + c$ is a perfect square number
Find the minimum value of $abc$ if triple $(a, b, c)$ is [i]brilliant[/i].
The sequence $(x_n)_{n\geqslant 0}$ is defined as such: $x_0=1, x_1=2$ and $x_{n+1}=4x_n-x_{n-1}$, for all $n\geqslant 1$. Determine all the terms of the sequence which are perfect squares.
[i]George Stoica, Canada[/i]
Find all integers $n$ for which $n(n + 2010)$ is a perfect square.