20

Part of 2011 Purple Comet Problems

Problems(2)

Purple Comet 2011 HS Problem 20

Source:

A

B

are the endpoints of a diameter of a circle with center

C

D

E

lie on the same diameter so that

C

bisects segment

\overline{DE}

F

be a randomly chosen point within the circle. The probability that

\triangle DEF

has a perimeter less than the length of the diameter of the circle is

\tfrac{17}{128}

. There are relatively prime positive integers m and n so that the ratio of

DE

AB

\tfrac{m}{n}.

m + n

probabilitygeometryperimeterrationumber theoryrelatively prime

Purple Comet 2011 MS Problem 20

Source:

V

be the set of vertices of a regular

25

sided polygon with center at point

C.

How many triangles have vertices in

V

and contain the point

C

in the interior of the triangle?