Problems

Which triangle has the largest area? The dots form a regular grid.

What is the ratio between the red and blue area? All shapes are semicircles.

In a parallelogram $ABCD$, point $E$ belongs to the side $CD$ and point $F$ belongs to the side $BC$. Show that the total red area is the same as the total blue area:

The figure below is a regular pentagram. What is larger, the black area or the blue area?

A circle was inscribed in a square, and another square was inscribed in the circle. Which area is larger, the blue or the orange one?

In a square, the midpoints of its sides were marked and some segments were drawn. There is another square formed in the centre. Find its area, if the side of the square has length $10$.

In a parallelogram $ABCD$, point $E$ belongs to the side $AB$, point $F$ belongs to the side $CD$ and point $G$ belongs to the side $AD$. What is more, the marked red segments $AE$ and $CF$ have equal lengths. Prove that the total grey area is equal to the total black area.

On the second day Robinson Crusoe stretched the rope between two pegs, put a ring on the rope, and tied the goat with another rope to the ring. What shape did the goat graze in this case?

On the third day Robinson Crusoe put two pegs again, and decided not to stretch the rope, but to tie the goat with two loose ropes of different lengths to those pegs. What shape did the goat graze on the third day?