Problems

In a regular hexagon, some diagonals were drawn. Find the area of the red region, if the total area of the hexagon is $72$.

Three semicircles are drawn on the sides of the triangle $ABC$ with sides $AB=3$, $AC=4$, $BC=5$ as diameters. Find the area of the red part.

$20$ birds fly into a photographer’s studio: $8$ starlings, $7$ wagtails and $5$ woodpeckers. Each time the photographer presses the shutter to take a photograph, one of the birds flies away and does not come back. How many photographs can the photographer take to be sure that at the end there will be no fewer than $5$ birds of one species and no less than 3 of another species remaining in the studio.

Prove that among $11$ different infinite decimal fractions, you can choose two fractions which coincide in an infinite number of digits.

A convex polygon on the plane contains at least $m^2+1$ points with integer coordinates. Prove that it contains $m+1$ points with integers coordinates that lie on the same line.

Suppose a football team scores at least one goal in each of the $20$ consecutive games. If it scores a total of $30$ goals in those $20$ games, prove that in some sequence of consecutive games it scores exactly $9$ goals total.

On the diagram below the line $BD$ is the bisector of the angle $\mathrm{\angle }ABC$ in the triangle $ABC$. A line through the vertex $C$ parallel to the line $BD$ intersects the continuation of the side $AB$ at the point $E$. Find the angles of the triangle $BCE$ triangle if $\mathrm{\angle }ABC={110}^{\circ }$.

Definition A set is a collection of elements, containing only one copy of each element. The elements are not ordered, nor they are governed by any rule. We consider an empty set as a set too.
There is a set $C$ consisting of $n$ elements. How many sets can be constructed using the elements of $C$?

Given a natural number $n$ you are allowed to perform two operations: "double up", namely get $2n$ from $n$, and "increase by $1$", i.e. to get $n+1$ from $n$. Find the smallest amount of operations one needs to perform to get the number $n$ from $1$.

In a certain state, there are three types of citizens:

• A fool considers everyone a fool and themselves smart;

• A modest clever person knows truth about everyone’s intellectual abilities and consider themselves a fool;

• A confident clever person knows about everyone intellectual abilities correctly and consider themselves smart.

There are $200$ deputies in the High Government. The Prime Minister conducted an anonymous survey of High Government members, asking how many smart people are there in the High Government. After reading everyone’s response he could not find out the number of smart people. But then the only member who did not participate in the survey returned from the trip. They filled out a questionnaire about the entire Government including themselves and after reading it the Prime Minister understood everything. How many smart could there be in the High Government (including the traveller)?