Problems

15 MPs take part in a debate. During the debate, each one criticises exactly $k$ of the 14 other contributors. For what minimum value of $k$ is it possible to definitively state that there will be two MPs who have criticised one another?

In the number $1234096\dots$ each digit, starting with the 5th digit, is equal to the final digit of the sum of the previous 4 digits. Will the digits 8123 ever occur in a row in this number?

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?

There are 30 ministers in a parliament. Each two of them are either friends or enemies, and each is friends with exactly six others. Every three ministers form a committee. Find the total number of committees in which all three members are friends or all three are enemies.

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?

7 natural numbers are written around the edges of a circle. It is known that in each pair of adjacent numbers one is divisible by the other. Prove that there will be another pair of numbers that are not adjacent that share this property.

On a function $f(x)$, defined on the entire real line, it is known that for any $a>1$ the function $f(x)+f(ax)$ is continuous on the whole line. Prove that $f(x)$ is also continuous on the whole line.

There are 7 points placed inside a regular hexagon of side length 1 unit. Prove that among the points there are two which are less than 1 unit apart.

Is it possible to transport 50 stone blocks, whose masses are equal to $370,372,\dots ,468$ kg, from a quarry on seven 3-tonne trucks?