Circle Second (Lesson3) – We Solve Problems

#### Circle Second (Lesson3)

Circle name: Circle Second
Lesson name: Lesson3
Starts at : 14.05.2020 09:00

Problems:

#### Examples and counterexamples. Constructive proofs , Pigeonhole principle (other) , Theory of algorithms (other)14-17

The function F is given on the whole real axis, and for each x the equality holds: F $(x + 1)$ F $(x)$ + F $(x + 1)$ + 1 = 0.
Prove that the function F can not be continuous.

#### Dissections (other)11-13

Two play a game on a chessboard 8 × 8. The player who makes the first move puts a knight on the board. Then they take turns moving it $($ according to the usual rules $)$, whilst you can not put the knight on a cell which he already visited. The loser is one who has nowhere to go. Who wins with the right strategy – the first player or his partner?

#### Pigeonhole principle (other)13-15

Prove that in any group of friends there will be two people who have the same number of friends.

#### Number tables and its properties , Pigeonhole principle (other)13-14

Is it possible to fill an nxn table with the numbers $-1, 0, 1$, such that the sums of all the rows, columns, and diagonals are unique?

#### Linear inequalites and systems of inequalities , Mathematical logic (other)13-15

On the Island of Bad Luck there are only knights who always tell the truth, and liars who always lie. In the government of the island there are 101 ministers. In order to reduce the budget, it was decided to reduce the number of ministers by 1. But each of the ministers said that if he was to be removed from the government, then the majority of the remaining ministers would be liars. How many knights and how many liars are there in the government?

#### Polygons (other)13-15

Arrows are placed on the sides of a polygon.
Prove that the number of vertices in which two arrows converge is equal to the number of vertices from which two arrows emerge.

#### Pigeonhole principle (other)15-17

Prove that there exist numbers, that can be presented in no fewer than 100 ways in the form of a summation of 20001 terms, each of which is the 2000th power of a whole number

#### Functions of one variable. Continuity , Iterations16-17

Let f $($x$)$ be a polynomial about which it is known that the equation f $($x$)$ = x has no roots. Prove that then the equation f $($f $($x$)$$)$ = x does not have any roots.

#### Theory of algorithms (other)13-14

A journalist came to a company which had N people. He knows that this company has a person Z, who knows all the other members of the company, but nobody knows him. A journalist can address each member of the company with the question: “Do you know such and such?” Find the smallest number of questions sufficient to surely find Z. $($Everyone answers the questions truthfully. One person can be asked more than one question.$)$

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