The last three digits of $ N$ are $ x25$. For how many values of $ x$ can $ N$ be the square of an integer?

Prove that the product of four consecutive integers plus $1$ is a perfect square.

Prove that there are do not exist natural numbers $k, m$ such that numbers $k^2+2m$, $m^2+2k$ to be squares of integers.

Show that for all positive integers $k$, there exists a positive integer n such that $n2^k -7$ is a perfect square.

Find the base of the number system less than $100$, in which $2101$ is a perfect square.

Determine all the possible positive integer $n,$ such that $3^n+n^2+2019$ is a perfect square.

Prove that an integer $n$ can be expressed as the sum of two squares if and only if $2n$ can be expressed as the sum of two squares.

For which integers $0 \le k \le 9$ do there exist positive integers $m$ and $n$ so that the number $3^m + 3^n + k$ is a perfect square?

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 $S(n)=n^{2}-14.$

The teacher wrote a non-zero natural number on the board. The teacher explained students that they can delete the number written on the board and can write a number instead naturally new, whenever they want, applying one of the following each time rules: 1) Instead of the current number $n$ write $3n + 13$ 2) Instead of the current number $n$ write the number $\sqrt{n}$, if $n$ is a perfect square . a) If the number $256$ was originally written on the board, is it possible that after a finite number of steps to get the number $55$ on the board? b) If the number $55$ was originally written on the board, is it possible that after a number finished the steps to get the number $256$ on the board?

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]

For which $n$ is $n^4+6n^3+11n^2+3n+31$ a perfect square?

For a positive integer $n$ denote $r(n)$ the number having the digits of $n$ in reverse order- for example, $r(2006) = 6002$. Prove that for any positive integers a and b the numbers $4a^2 + r(b)$ and $4b^2 + r(a)$ can not be simultaneously squares.

Let $P(x) =(x+1)^p (x-3)^q=x^n+a_1x^{n-1}+a_2x^{n-2}+...+a_{n-1}x+a_n$ where $p$ and $q$ are positive integers a) Given $a_1=a_2$, prove that $3n$ is a perfect square. b) Prove that there exist infinitely many pairs $(p, q)$ of positive integers p and q such that the equality $a_1=a_2$ is valid for the polynomial $P(x)$. (D. Bazylev)

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]