2

Part of 2014 Saint Petersburg Mathematical Olympiad

Problems(3)

Destroying the roads

Source: St Petersburg Olympiad 2014, Grade 11, P2

There are cities in country, and some cities are connected by roads. Not more than

100

roads go from every city. Set of roads is called as ideal if all roads in set have not common ends, and we can not add one more road in set without breaking this rule. Every day minister destroy one ideal set of roads. Prove, that he need not more than

199

days to destroy all roads in country.

Points on lines

Source: St Petersburg Olympiad 2014, Grade 10, P2

40

points on the two parallel lines. We divide it to pairs, such that line segments, that connects point in pair, do not intersect each other ( endpoint from one segment cannot lies on another segment). Prove, that number of ways to do it is less than

3^{39}

Source: St Petersburg Olympiad 2014, Grade 9, P2

ABC

(30,90)

. Circumcenter of

ABC

O

and circumradius is

R

K

is projection of

O

to angle bisector of

\angle B

M

AC

. It is known, that

2KM=R

\angle B

geometrycircumcircleangle bisector