Problems

Some inhabitants of the Island of Multi-coloured Frogs speak only the truth, and the rest always lie. Three islanders said:

Bree: There are no blue frogs on our island.

Kevin: Bree is a liar. She herself is a blue frog!

Clara: Of course, Bree is a liar. But she’s a red frog.

Are there any blue frogs on this island?

When cleaning her children’s room, a mother found $9$ socks. In a group of any $4$ of the socks at least two belonged to the same child. In a group of any $5$ of the socks no more than $3$ had the same owner. How many children are there in the room and how many socks belong to each child?

Let $x_1,x_2,\dots ,x_n$ be some numbers belonging to the interval $[0,1]$. Prove that on this segment there is a number $x$ such that $$\frac{1}{n}(|x-x_1|+|x-x_2|+\cdots +|x-x_n|)=1/2.$$

In the family of happy gnomes there is a father, a mother and a child. The names of the family members: Alex, Charlie and Jo. At the dinner table two gnomes made two statements.

Charlie said: “Alex and Jo are of different genders. Alex and Charlie are my parents”.

Alex said: “I am Jo’s father. I am the daughter of Charlie”.

Who is who? That is, what is the name of the father, the mother and the child, if it is known that each gnome lied once, and each told the truth once.

On the first day of school, in all three of the first year classes (1A, 1B, 1C), there were three lessons: Maths, French and Biology. Two classes cannot have the same lesson at the same time. 1B’s first lesson was Maths. The Biology teacher praised the students in 1B: “You have even better marks than 1A”. 1A’s second lesson was not French. Which class’s last lesson was Biology?

A bag contains balls of two different colours – black and white. What is the minimum number of balls you need to remove, without looking, to guarantee that within the removed balls at least two are the same colour.

A forest contains a million fir trees. It is known that any given tree has at most 600,000 needles. Prove that there will be two trees with the same number of needles.

You are given 12 different whole numbers. Prove that it is possible to choose two of these whose difference is divisible by 11.

A supermarket received a delivery of 25 crates of apples of 3 different types; each crate contains only one type of apple. Prove that there are at least 9 crates of apples of the same sort in the delivery.

In Scotland there are $m$ football teams containing 11 players each. All of the players met at the airport in order to travel to England for a match. The plane made 10 journeys from Scotland to England, carrying 10 passengers each time. One player also flew to the location of the match by helicopter. Prove that at least one team made it in its entirety to the other country to play the match.