Find the maximal $x$ such that the expression $4^{27} + 4^{1000} + 4^x$ is the exact square.

Let $n$ be a given positive integer. Prove that there is no positive divisor $d$ of $2n^2$ such that $d^2n^2+d^3$ is a square of an integer.

* The quadratic expression $ax^2 + bx + c$ is a square (of an integer) for any integer $x$. Prove that $ax^2 + bx + c = (dx + e)^2$ for some integers d and e.

A positive integer is called a [b]double number[/b] if its decimal representation consists of a block of digits, not commencing with 0, followed immediately by an identical block. So, for instance, 360360 is a double number, but 36036 is not. Show that there are infinitely many double numbers which are perfect squares.

Is it possible to represent the number $1986$ as the sum of squares of $6$ odd integers?

A positive integer $A_k...A_1A_0$ is called monotonic if $A_k \le ..\le A_1 \le A_0$. Show that for any $n \in N$ there is a monotonic perfect square with $n$ digits.

Find all positive integers $x$, $y$ such that $2^x+5^y+2$ is a perfect square.

Let $n$ be an integer greater than $1$, such that $3n + 1$ is a perfect square. Prove that $n + 1$ can be expressed as a sum of three perfect squares.

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

Can we find positive integers $a$ and $b$ such that both $(a^2 + b)$ and $(b^2 + a)$ are perfect squares?

Prove that $S_1 = (n + 1)^2 + (n + 2)^2 +...+ (n + 5)^2$ is divisible by $5$ for every $n$. Prove that for no $n$: $\sum_{\ell=1}^5 (n+\ell)^2$ is a perfect square. Let $S_2=(n + 6)^2 + (n + 7)^2 + ... + (n + 10)^2$. Prove that $S_1 \cdot S_2$ is divisible by $150$.

Find all positive integers $n$ such that $n^4 - n^3 + 3n^2 + 5$ is a perfect square.

Let $n > 1$ be a positive integer and $p$ a prime number such that $n | (p - 1) $and $p | (n^6 - 1)$. Prove that at least one of the numbers $p- n$ and $p + n$ is a perfect square.

Determine all integers $x$ such that $2x^2-x-36$ is a perfect square of a prime.

In the square of an integer $ a$, the tens’ digit is $7$. What is the units’ digit of $a^2$?

Find all positive integers $n$ for which $1 - 5^n + 5^{2n+1}$ is a perfect square.

Find the least possible number of elements which can be deleted from the set $\{1,2,...,20\}$ so that the sum of no two different remaining numbers is not a perfect square. N. Sedrakian , I.Voronovich

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].

Sabine has a very large collection of shells. She decides to give part of her collection to her sister. On the first day, she lines up all her shells. She takes the shells that are in a position that is a perfect square (the first, fourth, ninth, sixteenth, etc. shell), and gives them to her sister. On the second day, she lines up her remaining shells. Again, she takes the shells that are in a position that is a perfect square, and gives them to her sister. She repeats this process every day. The $27$th day is the first day that she ends up with fewer than $1000$ shells. The $28$th day she ends up with a number of shells that is a perfect square for the tenth time. What are the possible numbers of shells that Sabine could have had in the very beginning?

Find the sum of all non-zero digits that can repeat at the end of a perfect square. (For example, if $811$ were a perfect square, $1$ would be one of these non-zero digits.)

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.

Prove that there exists an infinite sequence of perfect squares with the following properties: (i) The arithmetic mean of any two consecutive terms is a perfect square, (ii) Every two consecutive terms are coprime, (iii) The sequence is strictly increasing.

Find all two-digit numbers $\overline {ab}$ such that $\overline {ab} + \overline {ba}$ is a perfect square.

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?

Prove that among any nine divisors of $30^{2010}$ there are two whose product is a perfect square.