Problems

Problems often involve a protagonist, a quest and a story. In combinatorics, stories can help us prove identities and formulas, that would be difficult to prove otherwise. Here, you can write your own story, which will show that the following statement is always true:

The number of ways we can choose \(k\) out of \(n\) items is equal to the number of ways we can choose \(k\) out of \(n-1\) objects PLUS the number of ways in which we can choose \(k-1\) out of \(n-1\) objects.

Can you cover a \(10 \times 10\) board using only \(T\)-shaped tetraminoes?

Can you cover a \(10 \times 10\) square with \(1 \times 4\) rectangles?

Two opposite corners were removed from an \(8 \times 8\) chessboard. Can you cover this chessboard with \(1 \times 2\) rectangular blocks?

One small square of a \(10 \times 10\) square was removed. Can you cover the rest of it with 3-square \(L\)-shaped blocks?

A \(7 \times 7\) square was tiled using \(1 \times 3\) rectangular blocks. One of the squares has not been covered. Which one can it be?

Can you cover a \(13 \times 13\) square using two types of blocks: \(2 \times 2\) squares and \(3 \times 3\) squares?

Inside a square with side 1 there are several circles, the sum of the radii of which is 0.51. Prove that there is a line that is parallel to one side of the square and that intersects at least 2 circles.

The cells of a \(15 \times 15\) square table are painted red, blue and green. Prove that there are two lines which at least have the same number of cells of one colour.

Cutting into four parts. Cut each of the figures below into four equal parts (you can cut along the sides and diagonals of cells).