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# Problem Solving Tactics

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.

# Problem Solving Tactics: Level 3 Challenges

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?

There are 100 runners, each given a distinct bib labeled 1 to 100. What is the most number of runners that we could arrange in a circle, such that the product of the numbers on the bibs of any 2 neighboring runners, is less than 1000?

Some unit squares of a $$2013 \times 2013$$ grid are marked so that any $$19 \times 19$$ subgrid has at least $$21$$ marked unit squares. What is the minimal possible number of marked unit squares?

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.

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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