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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.

Challenge Quizzes

Level 3

         

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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