Problems

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.

The prime factorization of the number \(b\) is \(2 \times 5^2 \times 7 \times 13^2 \times 17\). The prime factorization of the number \(c\) is \(2^2 \times 5 \times 7^2 \times 13\). Is the first number divisible by the second one? Is the product of these two numbers, \(b \times c\), divisible by \(49000\)?

Determine all prime numbers \(p\) such that \(5p+1\) is also prime.

On the diagram below \(AD\) is the bisector of the triangle \(ABC\). The point \(E\) lies on the side \(AB\), with \(AE = ED\). Prove that the lines \(AC\) and \(DE\) are parallel.

On the diagram below the line \(BD\) is the bisector of the angle \(\angle ABC\) 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 \(\angle ABC = 110^{\circ}\).

How many five-digit numbers are there which are written in the same from left to right and from right to left? For example the numbers \(54345\) and \(12321\) satisfy the condition, but the numbers \(23423\) and \(56789\) do not.

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 \(C\)?

Does there exist a power of \(3\) that ends in \(0001\)?

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)?