Problems
In a trapezium \(ABCD\), the side \(AB\) is parallel to the side \(CD\). Prove that the areas of triangles \(\triangle ABC\) and \(\triangle ABD\) are equal.
The triangle visible in the picture is equilateral. The hexagon inside is a regular hexagon. If the area of the whole big triangle is \(18\), find the area of the small blue triangle.
On the left there is a circle inscribed in a square with side \(1\). On the right there are \(1\)6 smaller, identical circles, which all together fit inside a square of side \(1\). Which area is greater, the yellow or the blue one?
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?
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.
On a certain planet the time zones can only differ by a multiple of \(1\) hour and their day is divided into hours in the same way Earth’s day is divided into hours. Show that if we pick \(25\) cities on that planet, some two cities will have the same local time.
The bag contains balls of two different colours: black and white. What is the smallest number of balls that need to be taken out of the bag blindly so that among them there are obviously two balls of the same colour?