Problems

There is a $5\times 9$ rectangle drawn on squared paper. In the lower left corner of the rectangle is a button. Kevin and Sophie take turns moving the button any number of squares either to the right or up. Kevin goes first. The winner is the one who places the button in upper right corner. Who would win, Kevin or Sophie, by using the right strategy?

The centres of all unit squares are marked in a $10\times 10$ chequered box (100 points in total). What is the smallest number of lines, that are not parallel to the sides of the square, that are needed to be drawn to erase all of the marked points?

All of the points with whole number co-ordinates in a plane are plotted in one of three colours; all three colours are present. Prove that there will always be possible to form a right-angle triangle from these points so that its vertices are of three different colours.

The city plan is a rectangle of $5\times 10$ cells. On the streets, a one-way traffic system is introduced: it is allowed to go only to the right and upwards. How many different routes lead from the bottom left corner to the upper right?

A ream of squared paper is shaded in two colours. Prove that there are two horizontal and two vertical lines, the points of intersection of which are shaded in the same colour.

An endless board is painted in three colours (each cell is painted in one of the colours). Prove that there are four cells of the same colour, located at the vertices of the rectangle with sides parallel to the side of one cell.

In the $6\times 7$ large rectangle shown below, how many rectangles are there in total formed by grid lines?

Prove that it’s impossible to cover a $4\times 9$ rectangle with $9$ ‘T’ tetrominoes (one copy seen in red).

One square is coloured red at random on an $8\times 8$ grid. Show that no matter where this red square is, you can cover the remaining $63$ squares with $21$ ‘L’ triominoes, with no gaps or overlaps.