Problems

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 is inscribed in a square, and another square is 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 connected to the vertices of the square. 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 $AD$. We know that the marked red segments $AE$ and $CF$ have equal lengths. Prove that the total grey area is equal to the total black area.

In a regular hexagon, some diagonals were drawn. Find the area of the red region, if the total area of the hexagon is $72$.

Three semicircles are drawn on the sides of the triangle $ABC$ with sides $AB=3$, $AC=4$, $BC=5$ as diameters. Find the area of the red part.

There are 8 students in an online chess club. Show that some two of them were born on the same day of the week.

Ramesh has an infinite number of red, blue and green socks in his drawer. How many socks does he need to pick from the drawer at random to guarantee he will have at least one pair of socks of one colour?

There are $6$ people playing a game online together. Among any $3$ people at least $2$ people know each other. Show that there is a group of $3$ people that all know each other.