Discrete Mathematics

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?

Satyen Nabar by Satyen Nabar

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?

Calvin Lin by Calvin Lin

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?

Zi Song Yeoh by Zi Song Yeoh

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

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

Calvin Lin by Calvin Lin

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.

Andrei Golovanov by Andrei Golovanov
×

Problem Loading...

Note Loading...

Set Loading...