Problems
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.
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?
Tom wants to go see a movie with his girlfriend Katie. Unfortunately the ticket selling service is broken and when you buy a ticket, you receive a ticket to a random movie that is being played that day. There are \(7\) movies being played on Friday. How many random tickets does Tom need to buy to guarantee that he and Katie will be able to go see a movie together?