3

Part of 2014 Stanford Mathematics Tournament

Problems(2)

SMT RMT 2014 Geometry #3

Source:

Spencer eats ice cream in a right circular cone with an opening of radius

5

and a height of

10

. If Spencer’s ice cream scoops are always perfectly spherical, compute the radius of the largest scoop he can get such that at least half of the scoop is contained within the cone.

SMT RMT 2014 Geometry Tiebreaker #3

Source:

O

has three chords,

AD

DF

EF

. Point E lies along the arc

AD

C

is the intersection of chords

AD

EF

B

lies on segment

AC

EB = EC = 8

AB = 6

BC = 10

CD = 9

DF

. https://cdn.artofproblemsolving.com/attachments/f/c/c36bff9ad04f13f7e227c57bddb53a0bfc0569.png