Problems

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 $$\frac{1}{n}(|x-x_1|+|x-x_2|+\cdots +|x-x_n|)=1/2.$$

On the selection to the government of the planet of liars and truth tellers $12$ candidates gave a speech about themselves. After a while, one said: “before me only once did someone lie” Another said: “And now-twice.” “And now – thrice” – said the third, and so on until the $12$th, who said: “And now $12$ times someone has lied.” Then the presenter interrupted the discussion. It turned out that at least one candidate correctly counted how many times someone had lied before him. So how many times have the candidates lied?

Two people play the following game. Each player in turn rubs out 9 numbers (at his choice) from the sequence $1,2,\dots ,100,101$. After eleven such deletions, 2 numbers will remain. The first player is awarded so many points, as is the difference between these remaining numbers. Prove that the first player can always score at least 55 points, no matter how played the second.

A six-digit phone number is given. How many seven-digit numbers are there from which one can obtain this six-digit number by deleting one digit?

There is a counter on the chessboard. Two in turn move the counter to an adjacent on one side cell. It is forbidden to put a counter on a cell, which it has already visited. The one who can not make the next turn loses. Who wins with the right strategy?

The city plan is a rectangle of $5\times 10$ cells. On the streets, a one-way traffic system is introduced: it is allowed to go only to the right and upwards. How many different routes lead from the bottom left corner to the upper right?

27 coins are given, of which one is a fake, and it is known that a counterfeit coin is lighter than a real one. How can the counterfeit coin be found from 3 weighings on the scales without weights?

Two people toss a coin: one tosses it 10 times, the other – 11 times. What is the probability that the second person’s coin showed heads more times than the first?

There are 30 students in the class. Prove that the probability that some two students have the same birthday is more than 50%.

Are there any irrational numbers $a$ and $b$ such that the degree of $a^b$ is a rational number?