Test1 (Lesson1) – We Solve Problems

Test1 (Lesson1)

Circle name: Test1
Lesson name: Lesson1
Starts at : 03.07.2020 09:00

Problems:

Game theory (other) , Theory of algorithms (other)10-12

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

Sequences15-18

For which natural $n$ does the number $\frac{n^2}{1,001^n}$ reach its maximum value?

Functional equations15-18

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.

Game theory (other) , Theory of algotithms10-13

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?

Counting in two ways , Exponential functions and logarithms (other) , Integer and fractional parts. Archimedean property16-18

Prove that for every natural number $n > 1$ the equality: $[n^{1 / 2}] + [n^{1/ 3}] + … + [n^{1 / n}] = [log_{2}n] + [log_{3}n] + … + [log_{n}n]$ is satisfied.