Main Circle (Introduction to maths) – We Solve Problems

#### Main Circle (Introduction to maths)

Circle name: Main Circle
Lesson name: Introduction to maths
Starts at : 23.07.2021 10:00

Problems:

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

A cat tries to catch a mouse in labyrinths A, B, and C. The cat walks first, beginning with the node marked with the letter “K”. Then the mouse $($ from the node “M”$)$ moves, then again the cat moves, etc. From any node the cat and mouse go to any adjacent node. If at some point the cat and mouse are in the same node, then the cat eats the mouse.
Can the cat catch the mouse in each of the cases A, B, C?
$\\$ $\\$
A                                                       B                                                          C

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

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

#### 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 players in turn increase a natural number in such a way that at each increase the difference between the new and old values of the number is greater than zero, but less than the old value. The initial value of the number is 2. The winner is the one who can create the number 1987. Who wins with the correct strategy: the first player or his partner?

#### Mathematical logic (other)11-13

In a vase, there is a bouquet of 7 white and blue lilac branches. It is known that 1$)$ at least one branch is white, 2$)$ out of any two branches, at least one is blue. How many white branches and how many blue are there in the bouquet?

#### 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

The vendor has a cup weighing scales with unequal shoulders and weights. First he weighs the goods on one cup, then on the other, and takes the average weight. Is he deceiving customers?

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