Problems

Prove that the product of

a) two odd numbers is odd;

b) an even number with any integer is even.

A road of length 1 km is lit with streetlights. Each streetlight illuminates a stretch of road of length 1 m. What is the maximum number of streetlights that there could be along the road, if it is known that when any single streetlight is extinguished the street will no longer be fully illuminated?

Several guests are sitting at a round table. Some of them are familiar with each other; mutually acquainted. All the acquaintances of any guest (counting himself) sit around the table at regular intervals. (For another person, these gaps may be different.) It is known that any two have at least one common acquaintance. Prove that all guests are familiar with each other.

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 that order 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?

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.

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.