Problems

In a basket there are 13 apples. There are scales, with which you can find out the total weight of any two apples. Think of a way to find out from 8 weighings the total weight of all the apples.

When boarding a plane, a line of $n$ passengers was formed, each of whom has a ticket for one of the $n$ places. The first in the line is a crazy old man. He runs onto the plane and sits down in a random place (perhaps, his own). Then passengers take turns to take their seats, and in the case that their place is already occupied, they sit randomly on one of the vacant seats. What is the probability that the last passenger will take his assigned seat?

We are given 101 natural numbers whose sum is equal to 200. Prove that we can always pick some of these numbers so that the sum of the picked numbers is 100.

Find all functions $f(x)$ defined for all real values of $x$ and satisfying the equation $2f(x)+f(1-x)=x^2$.

10 natural numbers are written on a blackboard. Prove that it is always possible to choose some of these numbers and write “$+$” or “$-$” between them so that the resulting algebraic sum is divisible by 1001.

On the board the number 1 is written. Two players in turn add any number from 1 to 5 to the number on the board and write down the total instead. The player who first makes the number thirty on the board wins. Specify a winning strategy for the second player.

There are two stacks of coins on a table: in one of them there are 30 coins, and in the other – 20. You can take any number of coins from one stack per move. The player who cannot make a move is the one that loses. Which player wins with the correct strategy?

There are 101 buttons of 11 different colours. Prove that amongst them there are either 11 buttons of the same colour, or 11 buttons of different colours.

A journalist came to a company which had $N$ people. He knows that this company has a person $Z$, who knows all the other members of the company, but nobody knows him. A journalist can address each member of the company with the question: “Do you know such and such?” Find the smallest number of questions sufficient to surely find $Z$. (Everyone answers the questions truthfully. One person can be asked more than one question.)

Let $f(x)$ be a polynomial about which it is known that the equation $f(x)=x$ has no roots. Prove that then the equation $f(f(x))=x$ does not have any roots.