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Difference between revisions of "2009 AIME II Problems/Problem 15"

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Let <math>\overline{MN}</math> 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=3/5. 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 chords AC and BC. The largest possible value of d can be written in the form r-s*sqrt (t), where r, s, and t are positive integers and t is not divisible by the square of any prime. Find r+s+t.
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Let <math>\overline{MN}</math> be a diameter of a circle with diameter 1. Let <math>A</math> and <math>B</math> be points on one of the semicircular arcs determined by <math>\overline{MN}</math> such that <math>A</math> is the midpoint of the semicircle and <math>MB= {3}over{5}</math>. Point <math>C</math> lies on the other semicircular arc. Let <math>d</math> be the length of the line segment whose endpoints are the intersections of diameter <math>\overline{MN}</math> with chords <math>\overline{AC}</math> and <math>\overline{BC}</math>. The largest possible value of <math>d</math> can be written in the form r-s*sqrt (t), where r, s, and t are positive integers and t is not divisible by the square of any prime. Find r+s+t.

Revision as of 05:22, 27 May 2009

Let $\overline{MN}$ be a diameter of a circle with diameter 1. Let $A$ and $B$ be points on one of the semicircular arcs determined by $\overline{MN}$ such that $A$ is the midpoint of the semicircle and $MB= {3}over{5}$. Point $C$ lies on the other semicircular arc. Let $d$ be the length of the line segment whose endpoints are the intersections of diameter $\overline{MN}$ with chords $\overline{AC}$ and $\overline{BC}$. The largest possible value of $d$ can be written in the form r-s*sqrt (t), where r, s, and t are positive integers and t is not divisible by the square of any prime. Find r+s+t.

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