10.9

Part of IV Soros Olympiad 1997 - 98 (Russia)

Problems(2)

max no of acute, 16 points (IV Soros Olympiad 1997-98 Correspondence 10.9)

Source:

16

points marked on the circle. Find the greatest possible number of acute triangles with vertices at the marked points.

geometrycombinatoricscombinatorial geometry

comp. geo with circumcircle and another circle

Source: IV Soros Olympiad 1997-98 R3 10.9 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

ABC

BC

a

, and the angles of the triangle adjacent to it are equal to

\alpha

\beta

. A circle passing through points

A

B

intersects lines

CA

CB

for second time at points

P

M

. It is known that straight line

RM

passes through the center of the circle circumscribed around

ABC

. Find the length of the segment

PM

geometrycircumcircle