Problems

Manraj wrote down a fraction, then he added 1 to the nominator and 100 to the denominator of the fraction. Could it be that the new fraction is bigger than the original one?

There are 10 strongman and 10 acrobats performing in a circus. At the beginning of the performance each strongman carried an acrobat to the arena, and at the end of the performance each acrobat carried a strongman offstage. It is known that each strongman carried an acrobat who weighed less than himself. Could it be that

(a) each acrobat carried a strongman lighter than himself? (b) there were nine acrobats each carrying a strongman lighter than himself?

The board of directors of a company consists of 4 people – one chairman and three ordinary members. The board has a meeting each month, where they decide on the amount of compensation each of them receives for serving on the board. According to the procedure the chairman proposes a new compensation scheme for all the members of the board, and all the members except the chairman vote for the new scheme subsequently. It is known that a member of the board votes for the scheme only if his/her compensation increases more or the same than everybody else’s, otherwise he/she votes against the scheme. The decisions are made according to majority rule. Can the chairman increase his/her compensation by 10 times, and simultaneously decrease every other member’s compensation by 10 times after several board meetings?

Is it possible to place several non-overlapping squares inside one big square with side length 1m if

(a) the sum of perimeters of smaller squares is equal to 100 m? (b) the sum of areas of smaller squares is equal to 100 m\(^2\)?

A labyrinth was drawn on a \(5\times 5\) grid square with an outer wall and an exit one cell wide, as well as with inner walls running along the grid lines. In the picture, we have hidden all the inner walls from you (We give you several copies to facilitate drawing) 
Please draw how the walls were arranged. Keep in mind that the numbers in the cells represent the smallest number of steps needed to exit the maze, starting from that cell. A step can be taken to any adjacent cell vertically or horizontally, but not diagonally (and only if there is no wall between them, of course).

Each integer point on the numerical axis is colored either white or black and the numbers \(2016\) and \(2017\) are colored differently. Prove that there are three identically colored integers which sum up to zero.

Prove that there are infinitely many prime numbers \(\{2,3,5,7,11,13...\}\).

The dragon locked six dwarves in the cave and said, "I have seven caps of the seven colors of the rainbow. Tomorrow morning I will blindfold you and put a cap on each of you, and hide one cap. Then I’ll take off the blindfolds, and you can see the caps on the heads of others, but not your own and I won’t let you talk any more. After that, everyone will secretly tell me the color of the hidden cap. If at least three of you guess right, I’ll let you all go. If less than three guess correctly, I’ll eat you all for lunch." How can dwarves agree in advance to act in order to be saved?