Problems

During the ball every young man danced the waltz with a girl, who was either more beautiful than the one he danced with during the previous dance, or more intelligent, but most of the men (at least 80%) – with a girl who was at the same time more beautiful and more intelligent. Could this happen? (There was an equal number of boys and girls at the ball.)

On a plane there is a square, and invisible ink is dotted at a point $P$. A person with special glasses can see the spot. If we draw a straight line, then the person will answer the question of on which side of the line does $P$ lie (if $P$ lies on the line, then he says that $P$ lies on the line).

What is the smallest number of such questions you need to ask to find out if the point $P$ is inside the square?

Ten little circles are drawn on a squared board $4\times 4$.

Cut the board into identical parts in such a way that each part contains 1, 2, 3, and 4 drawn circles correspondingly.

Philip and Denis cut a watermelon into four parts. When they finished eating watermelon (they ate the whole thing), they discovered that there were five watermelon rinds left. How is it possible, if no rind was cut after the initial cutting?

Cut a square into a heptagon (7 sides) and an octagon (8 sides) in such a way, that for every side of an octagon there exists an equal side belonging to the heptagon.

Cut a rectangle into two identical pentagons.

(a) Cut the rectangle into two identical quadrilaterals.

(b) Cut the rectangle into two identical hexagons.

(c) Cut the rectangle into two identical heptagons.

a) You have a $10\times 20$ chocolate bar and 19 friends. Since you are good at maths they ask you to split this bar into 19 pieces (always breaking along the lines between squares). All the pieces have to be of a rectangular shape. Your friends don’t really care how much they will get, they just want to be special, so you need to split the bar in such way that no two pieces are the same.

(b) The friends are quite impressed by your problem solving skills. But one of them is not that happy with the fact you didn’t get a single piece of the chocolate bar. He thinks you might feel that you are too special, therefore he convinces the others that you should get another $10\times 20$ chocolate bar and now split it into 20 different pieces, all of rectangular shapes (and still you need to break along the lines between squares). Can you do it now?

Can one cut a square into (a) one 30-gon and five pentagons? (b) one 33-gon and three 10-gons?

After having lots of practice with cutting different hexagons with a single cut Jennifer thinks she found a special one. She found a hexagon which cannot be cut into two quadrilaterals. Provide an example of such a hexagon.