Problems

A square of side 15 contains 20 non-overlapping unit squares. Prove that it is possible to place a circle of radius 1 inside the large square, so that it does not overlap with any of the unit squares.

Kate and Gina agreed to meet at the underground in the first hour of the afternoon. Kate comes to the meeting place between noon and one o’clock in the afternoon, waits for 10 minutes and then leaves. Gina does the same.

a) What is the probability that they will meet?

b) How will the probability of a meeting change if Gina decides to come earlier than half past twelve, and Kate still decides to come between noon and one o’clock?

c) How will the probability of a meeting change if Gina decides to come at an arbitrary time between 12:00 and 12:50, and Kate still comes between 12:00 and 13:00?

In a pentagon \(ABCDE\), diagonal \(AD\) is parallel to the side \(BC\) and the diagonal \(CE\) is parallel to the side \(AB\). Show that the areas of the triangles \(\triangle ABE\) and \(\triangle BCD\) are the same.

The area of the red triangle is \(25\) and the area of the orange triangle is \(49\). What is the area of the trapezium \(ABCD\)?

Prove that the relation between areas of two similar polygons equals to the square of their similarity coefficient.