Prove there are no integer solutions for the equation \(3x^2 + 2 = y^2\).
Can seven phones be connected with wires so that each phone is connected to exactly three others?
a) Can 4 points be placed on a plane so that each of them is connected by segments with three points (without intersections)?
b) Can 6 points be placed on a plane and connected by non-intersecting segments so that exactly 4 segments emerge from each point?
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.
During the election for the government of the planet of Liars and
Truth-Tellers, \(12\) candidates each
gave a short speech about themselves.
After everyone had spoken, one alien said: “So far, only one lie has
been told today.”
Then another said: “And now two have been said so far.”
The third said: “And now three lies have been told so far,” and so on —
until the twelfth alien said: “And now twelve lies have been told so
far.”
It turned out that at least one candidate had correctly counted how many
lies had been told before their own statement.
How many lies were said that day in total?
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.