MathDB

One eastern country was ruled by an old Shah. The population of the country consisted of inhabitants and satraps. Each resident had his own place of residence (place of registration). Satraps moved around the country and carried out the decrees of the Shah. One day the Shah issued a decree containing the following points:
1) Some residents are bandits. 2) Every bandit must be destroyed. 3) Together with the bandit, all those residents who are located closer to the bandit than the Shah (in other words, than the location of the Shah’s palace) must be destroyed.
Finding out which of the residents was a bandit was entrusted to the Shah's adviser, known for his connections with one hostile state. Prove that:
a) if the country in question is on a plane, then the adviser has the opportunity to declare no more than six inhabitants bandits in such a way that all inhabitants of the country must be destroyed in accordance with the decrees;
b) if the country is located on a sphere, then you can get by with five bandits.

All sides of triangle

ABC

are different. On rays

B A

and

C A

the segments

B K

and

CM

are laid out, equal to side

BC

. Let us denote by

x

the length of the segment

KM

. In the same way, by plotting the side

AC

on the rays

AB

and

CB

from

A

and

C

, we obtain a segment of length

y

, and by plotting the side AB on the rays

AC

and

BC

, we obtain a segment of length

z

. a) Prove that a triangle can be formed from the segments

x

y

and

z

, and this triangle is similar to triangle

ABC

. b) Find the radius of the circumcircle of a triangle with sides

x

y

and

z

, if the radii of the circumscribed and inscribed circles of triangle

ABC

are equal to

R

and

r

respectively.

11.10

Problems(2)