Determine whether there exists an infinite sequence of nonzero digits $a_1 , a_2 , a_3 , \cdots $ and a positive integer $N$ such that for every integer $k > N$, the number $\overline{a_k a_{k-1}\cdots a_1 }$ is a perfect square.

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.)

(a) Let $m$ and $n$ be positive integers and $p$ a positive rational number, with $m > n$, such that $\sqrt{m} -\sqrt{n}= p$. Prove that $m$ and $n$ are perfect squares. (b) Find all four-digit numbers $\overline{abcd}$, where each letter $a, b, c$ and $d$ represents a digit, such that $\sqrt{\overline{abcd}} -\sqrt{\overline{acd}}= \overline{bb}$.

Show that, if $m$ and $n$ are non-zero integers of like parity, and $n^2 -1$ is divisible by $m^2 - n^2 + 1$, then $m^2 - n^2 + 1$ is the square of an integer. Amer. Math. Monthly

Let the sequence $\{a_n\}$ satisfy $a_{n+1}=a_n^3+103,n=1,2,...$. Prove that at most one integer $n$ such that $a_n$ is a perfect square.

Let $n \ge 2003$ be a positive integer such that the number $1 + 2003n$ is a perfect square. Prove that the number $n + 1$ is equal to the sum of $2003$ positive perfect squares.

Find all prime numbers $p$ such that $p^3-4p+9$ is a perfect square.

Can a number of the form $44\dots 41$, with an odd number of decimal digits $4$ followed by a digit $1$, be a perfect square?

Prove there aren't positive integers $a, b, c, d$ forming an arithmetic progression such that $ ab + 1, ac + 1, ad + 1, bc + 1, bd + 1, cd + 1 $ are all perfect squares.

A [i]permutation[/i] of the set of positive integers $[n] = \{1, 2, . . . , n\}$ is a sequence $(a_1 , a_2 , \ldots, a_n ) $ such that each element of $[n]$ appears precisely one time as a term of the sequence. For example, $(3, 5, 1, 2, 4)$ is a permutation of $[5]$. Let $P (n)$ be the number of permutations of $[n]$ for which $ka_k$ is a perfect square for all $1 \leq k \leq n$. Find with proof the smallest $n$ such that $P (n)$ is a multiple of $2010$.

Find all positive integers $n$ such that $7^n + 147$ is a perfect square.

Solve $ x^3-y^3=2xy+7 $ in integers.

An integer $c$ is [i]square-friendly[/i] if it has the following property: For every integer $m$, the number $m^2+18m+c$ is a perfect square. (A perfect square is a number of the form $n^2$, where $n$ is an integer. For example, $49 = 7^2$ is a perfect square while $46$ is not a perfect square. Further, as an example, $6$ is not [i]square-friendly[/i] because for $m = 2$, we have $(2)2 +(18)(2)+6 = 46$, and $46$ is not a perfect square.) In fact, exactly one square-friendly integer exists. Show that this is the case by doing the following: (a) Find a square-friendly integer, and prove that it is square-friendly. (b) Prove that there cannot be two different square-friendly integers.

Let us call a set of positive integers nice if the number of its elements equals to the average of its numbers. Call a positive integer $n$ an [i]amazing[/i] number if the set $\{1, 2 , . . . , n\}$ can be partitioned into nice subsets. a) Prove that every perfect square is amazing. b) Show that there are infinitely many positive integers which are not amazing.

Let $n$ be positive integer such that $\sqrt{1 + 12n^2}$ is an integer. Prove that $2 + 2\sqrt{1 + 12n^2}$ is the square of an integer.

Suppose that $a$ and $ b$ are positive integers such that $$c = a +\frac{b}{a} -\frac{1}{b}$$ is an integer. Prove that $c$ is a perfect square.

Natural numbers $ x, y, z $ whose greatest common divisor is equal to 1 satisfy the equation $$\frac{1}{x} + \frac{1}{y} = \frac{1}{z}$$ Prove that $ x + y $ is the square of a natural number.

An infinite arithmetic progression whose terms are positive integers contains the square of an integer and the cube of an integer. Show that it contains the sixth power of an integer.

Find all integers $k{}$ for which there are infinitely many triples of integers $(a,b,c)$ such that \[(a^2-k)(b^2-k)=c^2-k.\]

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 product $1\times 2\times 3\times ...\times n$ is written on the board. For what integers $n \ge 2$, we can add exclamation marks to some factors to convert them into factorials, in such a way that the final product can be a perfect square?

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.

Find all positive integer pairs $(a,b)$ such that, $$\frac{10^{a!} - 3^b +1}{2^a}$$ is a perfect square.

Find all positive real numbers $c$ such that there are infinitely many pairs of positive integers $(n,m)$ satisfying the following conditions: $n \ge m+c\sqrt{m - 1}+1$ and among numbers $n. n+1,.... 2n-m$ there is no square of an integer. (Slovakia)

Determine all positive integers $n$ such that $2022 + 3^n$ is a perfect square.