A ream of squared paper is shaded in two colours. Prove that there are two horizontal and two vertical lines, the points of intersection of which are shaded in the same colour.
There are 5 points inside an equilateral triangle with side of length 1. Prove that the distance between some two of them is less than 0.5.
A \(3\times 4\) rectangle contains 6 points. Prove that amongst them there will be two points, such that the distance between them is no greater than \(\sqrt5\).
There are 25 points on a plane, and among any three of them there can be found two points with a distance between them of less than 1. Prove that there is a circle of radius 1 containing at least 13 of these points.
What is the minimum number of points necessary to mark inside a convex \(n\)-gon, so that at least one marked point always lies inside any triangle whose vertices are the vertices of the polygon?
a) In Wonderland, there are three cities \(A\), \(B\) and \(C\). 6 roads lead from city \(A\) to city \(B\), and 4 roads lead from city \(B\) to city \(C\). How many ways can you travel from \(A\) to \(C\)?
b) In Wonderland, another city \(D\) was built as well as several new roads – two from \(A\) to \(D\) and two from \(D\) to \(C\). In how many ways can you now get from city \(A\) to city \(C\)?
How many distinct seven-digit numbers exist? It is assumed that the first digit cannot be zero.
A car registration number consists of three letters of the Russian alphabet (that is, 30 letters are used) and three digits: first we have a letter, then three digits followed by two more letters. How many different car registration numbers are there?
We call a natural number “fancy”, if it is made up only of odd digits. How many four-digit “fancy” numbers are there?
A sack contains 70 marbles, 20 red, 20 blue, 20 yellow, and the rest black or white. What is the smallest number of marbles that need to be removed from the sack, without looking, in order for there to be no less than 10 marbles of the same colour among the removed marbles.