Problems

Age
Difficulty
Found: 2016

The symmetry axis of the polygon intersects its sides at points \(A\) and \(B\). Prove that the point \(A\) is either the vertex of the polygon or the middle of the side perpendicular to the axis of symmetry.

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.

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 \(n\)-sided polygon, so that at least one marked point always lies inside any triangle whose vertices are shared with those 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.