Problems

In front of a gnome there lie three piles of diamonds: one with 17, one with 21 and one with 27 diamonds. In one of the piles lies one fake diamond. All the diamonds have the same appearance, and all real diamonds weigh the same, and the fake one differs in its weight. The gnome has a cup weighing scale without weights. The dwarf must find with one weighing a pile, in which all the diamonds are real. How should he do it?

Ladybirds gathered in a sunny clearing. If the ladybird has $6$ spots, then it always speaks the truth, and if it has $4$ spots, then it always lies. There are no other types of ladybirds in the meadow. The first ladybird said: “We each have the same number of spots on our backs.” The second one said: “Everyone has $30$ spots on their backs in total.” “No, we all have $26$ spots on their backs in total,” the third objected. “Of these three, exactly one told the truth,” – said each of the other ladybirds. How many ladybugs were gathered in the meadow?

We are given $n+1$ different natural numbers, which are less than $2n$ ($n>1$). Prove that among them there will always be three numbers, where the sum of two of them is equal to the third.

Thirty girls – 13 in red dresses and 17 in blue dresses – led a dance around the Christmas tree. Subsequently, each of them was asked if her neighbour on the right was in a blue dress. It turned out that those girls which answered correctly were only those who stood between two girls in dresses of the same color. How many girls could have said yes?

Here’s a rather simple rebus:

$EX$ is four times larger than $OJ$.

$AJ$ is four times larger than $OX$.

Find the sum of all four numbers.

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?