Problem Solving Tactics

Problem Solving Tactics: Level 3 Challenges


Consider the set

S={1,12,13,14,,1100}. S= \left \{ 1, \frac {1}{2}, \frac {1}{3}, \frac {1}{4},\cdots, \frac {1}{100} \right \}.

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

For example, if we choose the numbers 12\frac{1}{2} and 18\frac {1}{8}, we will replace them by 1116 \frac {11}{16} .

If we keep repeating this process until only 11 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 2013×20132013 \times 2013 grid are marked so that any 19×1919 \times 19 subgrid has at least 2121 marked unit squares. What is the minimal possible number of marked unit squares?

Find the largest positive integer n<100,n<100, such that there exists an arithmetic progression of positive integers a1,a2,...,ana_1,a_2,...,a_n with the following properties.

1) All numbers a2,a3,...,an1a_2,a_3,...,a_{n-1} are powers of positive integers, that is numbers of the form jk,j^k, where j1j\geq 1 and k2k\geq 2 are integers.

2) The numbers a1a_1 and ana_{n} are not powers of positive integers.

Sergei chooses two different natural numbers aa and bb. He writes four numbers in a notebook: aa, a+2a+2, bb and b+2b+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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