Write out in a row the numbers from \(1\) to \(9\) (every number once) so that every two consecutive numbers give a two-digit number that is divisible by \(7\) or by \(13\).
Several Top Secret Objects are connected by an underground railway in such a way that each Object is directly connected to no more than three others and from each Object one can reach any other Object by going and by changing no more than once. What is the maximum number of Top Secret Objects?
Prove that the sum of
a) any number of even numbers is even;
b) an even number of odd numbers is even;
c) an odd number of odd numbers is odd.
Prove that the product of
a) two odd numbers is odd;
b) an even number with any integer is even.
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.
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?
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.