Discrete Mathematics
# Problem Solving Tactics

Consider the set

$S= \left \{ 1, \frac {1}{2}, \frac {1}{3}, \frac {1}{4},\cdots, \frac {1}{100} \right \}.$

Choose any two numbers $x$ and $y,$ and replace them with $x+y+ xy.$

For example, if we choose the numbers $\frac{1}{2}$ and $\frac {1}{8}$, we will replace them by $\frac {11}{16}$.

If we keep repeating this process until only $1$ number remains, what is the final number?

Find the largest positive integer $n<100,$ such that there exists an arithmetic progression of positive integers $a_1,a_2,...,a_n$ with the following properties.

1) All numbers $a_2,a_3,...,a_{n-1}$ are powers of positive integers, that is numbers of the form $j^k,$ where $j\geq 1$ and $k\geq 2$ are integers.

2) The numbers $a_1$ and $a_{n}$ are not powers of positive integers.

$a$ and $b$. He writes four numbers in a notebook: $a$, $a+2$, $b$ and $b+2$.

Sergei chooses two different natural numbersHe then writes all six pairwise products of the numbers of notebook on the blackboard.

What is the maximum number of perfect squares on the blackboard?

**Assumption:** Natural numbers don't include zero.