On a plane, there are 1983 points and a circle of unit radius. Prove that there is a point on the circle, from which the sum of the distances to these points is no less than 1983.
What is the minimum number of lottery tickets for the Sport Lotto that it is necessary to buy in order to guarantee that at least one of the tickets will have one number correct. On any single ticket you can choose 6 of the available numbers 1 to 49.
There were seven boxes. In some of them, seven more boxes were placed inside (not nested in each other), etc. As a result, there are 10 non-empty boxes. How many boxes are there now in total?
Four aliens – Dopey, Sleepy, Happy, Moody from the planet of liars and truth tellers had a conversation: Dopey to Sleepy: “you are a liar”; Happy to Sleepy: “you are a liar”; Moody to Happy: “Yes, they are both liars,” (after a moment’s thought), “however, so are you.” Which of them is telling the truth?
A class contains 25 pupils. It is known that within any group of 3 pupils there are two friends. Prove that there is a pupil who has no fewer than 12 friends.
One corner square was cut from a chessboard. What is the smallest number of equal triangles that can be cut into this shape?
It is known that a certain polynomial at rational points takes rational values. Prove that all its coefficients are rational.
Reception pupil Peter knows only the number 1. Prove that he can write a number divisible by 1989.
A scone contains raisins and sultanas. Prove that inside the scone there will always be two points 1cm apart such that either both lie inside raisins, both inside sultanas, or both lie outside of either raisins or sultanas.
The order of books on a shelf is called wrong if no three adjacent books are arranged in order of height (either increasing or decreasing). How many wrong orders is it possible to construct from \(n\) books of different heights, if: a) \(n = 4\); b) \(n = 5\)?