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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?

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)?

There are 100 people standing in line, and one of them is Arthur. Everyone in the line is either a knight, who always tells the truth, or a liar who always lies. Everyone except Arthur said, "There are exactly two liars between Arthur and me." How many liars are there in this line, if it is known that Arthur is a knight?

A rectangular parallelepiped of the size m×n×k is divided into unit cubes. How many rectangular parallelepipeds are formed in total (including the original one)?

Winnie the Pooh has five friends, each of whom has pots of honey in their house: Tigger has 1 pot, Piglet has 2, Owl has 3, Eeyore has 4, and Rabbit has 5. Winnie the Pooh comes to visit each friend in turn, eats one pot of honey and takes the other pots with him. He came into the last house carrying 10 pots of honey. Whose house could Pooh have visited last?

In the Land of Linguists live m people, who have opportunity to speak n languages. Each person knows exactly three languages, and the sets of known languages may be different for different people. It is known that k is the maximum number of people, any two of whom can talk without interpreters. It turned out that 11nkm/2. Prove that then there are at least mn pairs of people in the country who will not be able to talk without interpreters.

Each integer on the number line is coloured either white or black. The numbers 2016 and 2017 are coloured differently. Prove that there are three identically coloured integers which sum to zero.

There are 100 non-zero numbers written in a circle. Between every two adjacent numbers, their product was written, and the previous numbers were erased. It turned out that the number of positive numbers after the operation coincides with the amount of positive numbers before. What is the minimum number of positive numbers that could have been written initially?