Problems

Among 4 people there are no three with the same name, the same middle name and the same surname, but any two people have either the same first name, middle name or surname. Can this be so?

I am going to convince you that all people have the same eye color! How? Well, notice that if there were only one person in the world, then my claim would be true. Now we will explain that if the claim is true when there are \(n\) people in the world, then it will also be true when there are \(n+1\) people. Therefore, it will be true regardless of the amount of people! (This kind of proof is called a proof by induction)

Let’s imagine that the claim is true when there are \(n\) people in the world. Now take any group of \(n+1\) people, and label them \(a_1,a_2,\dots,a_{n+1}\). Remove \(a_1\). The remaining people \(a_2,a_3,\cdots,a_{n+1}\) form a group of \(n\) people, so they must all have the same eye colour.

On the other hand, let’s remove \(a_{n+1}\). The remaining people \(a_1,a_2,\dots,a_{n}\) also form a group of \(n\) people, so they must again all have the same eye colour.

Since these two smaller groups overlap (they both contain \(a_2,a_3,\dots,a_{n-1}\)), everyone in the full group \(a_1,a_2,\dots,a_n\) has the same eye colour.

Nick writes the numbers \(1,2,\dots,33\), each exactly once, at the vertices of a polygon with \(33\) sides, in some order.

For each side of the polygon, his little sister Hannah writes down the sum of the two numbers at its ends. In total she writes down \(33\) numbers, one for each side.

It turns out that when read in order around the polygon, these \(33\) sums are \(33\) consecutive whole numbers.

Can you find an arrangement of the numbers written by Nick that makes this happen?

Is it possible to arrange the numbers \(1,\, 2,\, ...,\, 50\) at the vertices and middles of the sides of a regular \(25\)-gon so that the sum of the three numbers at the ends and in the middle of each side is the same for all sides?

Draw a shape that can both be cut into 4 copies of the figure on the left or alternatively into 5 copies of the figure on the right. (the figures can be rotated).

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.

A bag contains balls of two different colours: black and white. What is the smallest number of balls that needs to be taken out of the bag blindly so that among them there are obviously two balls of the same colour?