Circle Second (x) – We Solve Problems

#### Circle Second (x)

Circle name: Circle Second
Lesson name: x
Starts at : 06.05.2020 09:00

Problems:

#### Arithmetics. Mental maths , Mathematical logic (other)12-13

This problem is from Ancient Rome.
$\\$ A rich senator died, leaving his wife pregnant. After the senator’s death it was found out that he left a property of 210 talents (an Ancient Roman currency) in his will as follows: “In the case of the birth of a son, give the boy two thirds of my property (i.e. 140 talents) and the other third (i.e. 70 talents) to the mother. In the case of the birth of a daughter, give the girl one third of my property (i.e. 70 talents) and the other two thirds (i.e. 140 talents) to the mother.”
$\\$ The senator’s widow gave birth to twins: one boy and one girl. This possibility was not foreseen by the late senator. How can the property be divided between three inheritors so that it is as close as possible to the instructions of the will?

#### Theory of algorithms (other)11-13

Burbot-Liman. Find the numbers that, when substituted for letters instead of the letters in the expression NALIM × 4 = LIMAN, fulfill the given equality (different letters correspond to different numbers, but identical letters correspond to identical numbers)

#### Theory of algorithms (other)11-13

Decipher the following rebus. Despite the fact that only two figures are known here, and all the others are replaced by asterisks, the question can be restored.$\\$

#### Boundedness, monotonicity , Quadratic inequaities and systems of inequalities14-17

For which natural K does the number reach its maximum value?

#### Regular polygons , Symmetric strategies13-15

a) The vertices (corners) in a regular polygon with 10 sides are coloured black and white in an alternating fashion (i.e. one vertice is black, the next is white, etc). Two people play the following game. Each player in turn draws a line connecting two vertices of the same colour. These lines must not have common vertices (i.e. must not begin or end on the same dot as another line) with the lines already drawn. The winner of the game is the player who made the final move. Which player, the first or the second, would win if the right strategy is used?

b) The same problem, but for a regular polygon with 12 sides.

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