Found problems: 521
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]
Consider the equation $3y^4 + 4cy^3 + 2xy + 48 = 0$, where $c$ is an integer parameter. Determine all values of $c$ for which the number of integral solutions $(x,y)$ satisfying the conditions (i) and (ii) is maximal:
(i) $|x|$ is a square of an integer;
(ii) $y$ is a squarefree number.
Let $k$ be a positive integer. Show that exists positive integer $n$, such that sets $A = \{ 1^2, 2^2, 3^2, ...\}$ and $B = \{1^2 + n, 2^2 + n, 3^2 + n, ... \}$ have exactly $k$ common elements.
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?
For which integer \( h \), are there infinitely many positive integers \( n \) such that \( \lfloor \sqrt{h^2 + 1} \cdot n \rfloor \) is a perfect square? (Here \( \lfloor x \rfloor \) denotes the integer part of the real number \( x \)?
The $x$ and $y$ figures satisfy a condition: for every $n\ge1$ the number $$xx...x6yy...y4$$ ($n$ times $x$ and $n$ times $y$) is a perfect square. Find all possible $x$ and $y$.
Determine for what $n\ge 3$ integer numbers, it is possible to find positive integer numbers $a_1 < a_2 < ...< a_n$ such $\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}=1$ and $a_1 a_2\cdot\cdot\cdot a_n$ is a perfect square.
Let $a$ and $b$ be integers, where $b$ is not a perfect square. Prove that $x^2 + ax + b$ may be the square of an integer only for finite number of integer values of $x$.
(Martin Panák)
Find all integers $k$ such that both $k + 1$ and $16k + 1$ are perfect squares.
$n$ and $d$ are positive integers such that $d$ divides $2n^2$. Prove that $n^2 + d$ cannot be a square.
The first digit of a $6$-digit number is $5$. Is it true that it is always possible to write $6$ more digits to the right of this number so that the resulting $12$-digit number is a perfect square?
(A Tolpygo)
Find all positive integers $x$, $y$ such that $2^x+5^y+2$ is a perfect square.
Show that the product of three consecutive positive integers is never a perfect square.
How many natural numbers are there whose square is a thirty-digit number which has the following curious property: If that thirty-digit number is divided from left to right into three groups of ten digits, then the numbers given by the middle group and the right group formed numbers are both four times the number formed by the left group?
Let $a, b,c$ be positive integers with no common factor and satisfy the conditions $\frac1a +\frac1b=\frac1c$
Prove that $a + b$ is a square.
Prove that the set of all divisors of a positive integer which is not a perfect square can be divided into pairs so that in each pair, one number is divided by another.
Let $n$ be a $6$-digit number, perfect square and perfect cube, if $n -6$ is neither even nor multiple of $3$. Find $n$ .
For which positive integers $n \ge 2$ it is possible to represent the number $n^2$ as a sum of n distinct positive integers not exceeding $\frac{3n}{2}$ ?
Find unequal integers $m, n$ such that $mn + n$ and $mn + m$ are both squares. Can you find such integers between $988$ and $1991$?
Consider the three squares indicated in the figure. Show that if the lengths of the sides of the smaller square and the square greater are integers, then adding to the area of the smallest square the area of the inclined square, a perfect square is obtained.
[img]https://1.bp.blogspot.com/-B0QdvZIjOLw/X4URvs3C0ZI/AAAAAAAAMmw/S5zMpPBXBn8Jj39d-OZVtMRUDn3tXbyWgCLcBGAsYHQ/s0/2002%2Bportugal%2Bp5.png[/img]
A number $N$ that is not divisible by $81$ can be represented as a sum of squares of three integers divisible by $3$. Prove that it is also representable as the sum of the squares of three integers not divisible by $3$.
Prove that there exist infinitely many positive integers $n$ such that the greatest prime divisor of $n^2+1$ is less than $n \cdot \pi^{-2019}.$
For the natural numbers $x$ and $y$, $2x^2 + x = 3y^2 + y$ .
Prove that then $x-y$, $2x + 2y + 1$ and $3x + 3y + 1$ are perfect squares.
Find all primes $p$ such that $(p-3)^p+p^2$ is a perfect square.
Find all $3$-tuples $(a, b, c)$ of positive integers, with $a \geq b \geq c$, such that $a^2 + 3b$, $b^2 + 3c$, and $c^2 + 3a$ are all squares.