Problem:
Let MN be a diameter of a circle with diameter 1. Let A and B be points on one of the semicircular arcs determined by MN such that A is the midpoint of the semicircle and MB=53. Point C lies on the other semicircular arc. Let d be the length of the line segment whose endpoints are the intersections of diameter MN with the chords AC and BC. The largest possible value of d can be written in the form r−st, where r, s, and t are positive integers and t is not divisible by the square of any prime. Find r+s+t.
Solution:
Let [XYZ] represent the area of △XYZ.
Let BC and AC intersect MN at points P and Q respectively, and let CNCM=x. Then
NPMP=[BNC][BMC]=BN⋅CNsin∠BNCBM⋅CMsin∠BMC=43x
Because NP=1−MP, it follows that MP=3x+43x. Similarly,
NQMQ=[ANC][AMC]=AN⋅CNsin∠ANCAM⋅CMsin∠AMC=x
giving MQ=x+1x. The fact that MQ−MP=d implies that x+1x−3x+43x=d, or equivalently, 3dx2+(7d−1)x+4d=0. Because the discriminant of this equation, which is d2−14d+1, must be nonnegative and 0<d<1, the largest value of d is 7−43, and r+s+t=14.
