You are given 1002 different integers that are no greater than 2000. Prove that it is always possible to choose three of the given numbers so that the sum of two of them is equal to the third.
Will this still always be possible if we are given 1001 integers rather than 1002?
On a plane, six points are given so that no three of them lie on the same line. Each pair of points is connected by a blue or red segment.
Prove that among these points three such points can be chosen so that all sides of the triangle formed by them will be of the same colour.
On a plane \(n\) randomly placed lines are given. What is the number of triangles formed by them?
How many six-digit numbers exist, for which each succeeding number is smaller than the previous one?
Why are the equalities \(11^2 = 121\) and \(11^3 = 1331\) similar to the lines of Pascal’s triangle? What is \(11^4\) equal to?
Find \(m\) and \(n\) knowing the relation \(\binom{n+1}{m+1}: \binom{n+1}{m}:\binom{n+1}{m-1} = 5:5:3\).
In a box, there are 10 white and 15 black balls. Four balls are removed from the box. What is the probability that all of the removed balls will be white?
Write at random a two-digit number. What is the probability that the sum of the digits of this number is 5?
There are three boxes, in each of which there are balls numbered from 0 to 9. One ball is taken from each box. What is the probability that
a) three ones were taken out;
b) three equal numbers were taken out?
A player in the card game Preferans has 4 trumps, and the other 4 are in the hands of his two opponents. What is the probability that the trump cards are distributed a) \(2: 2\); b) \(3: 1\); c) \(4: 0\)?