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Learn about advanced problem-solving tactics such as Construction, the Extremal Principle, and the Invariant Principle, and you'll be crushing tricky problems in no time.

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?

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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.

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Sergei chooses two different natural numbers \(a\) and \(b\). He writes four numbers in a notebook: \(a\), \(a+2\), \(b\) and \(b+2\).

He 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.

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