2009 AIME II Problems/Problem 15
Let be a diameter of a circle with diameter 1. Let and be points on one of the semicircular arcs determined by such that is the midpoint of the semicircle and . Point lies on the other semicircular arc. Let be the length of the line segment whose endpoints are the intersections of diameter with chords and . The largest possible value of can be written in the form , where and are positive integers and is not divisible by the square of any prime. Find .
Solution 1 (Quick Calculus)
Let and . Further more let and . Angle chasing reveals and . Additionally and by the Pythagorean Theorem.
By the Angle Bisector Formula,
As we compute and , and finally . Taking the derivative of with respect to , we arrive at Clearly the maximum occurs when . Plugging this back in, using the fact that and , we get
Solution 2 (Projective)
Since , point lies between and on the semicircular arc. We will first compute the length of . By the law of cosines, , so . Then , so .
Let and , and let , , . Note thatthat is,orHence , and we also know . Now AM-GM givesThis gives the quadratic inequality , which solves asBut , so the greatest possible value of is . The answer is .
Solution 3 (Calculus)
Let be the center of the circle. Define , , and let and intersect at points and , respectively. We will express the length of as a function of and maximize that function in the interval .
Let be the foot of the perpendicular from to . We compute as follows.
(a) By the Extended Law of Sines in triangle , we have
(b) Note that and . Since and are similar right triangles, we have , and hence,
(c) We have and , and hence by the Law of Sines,
(d) Multiplying (a), (b), and (c), we have
which is a function of (and the constant ). Differentiating this with respect to yields
and the numerator of this is
which vanishes when . Therefore, the length of is maximized when , where is the value in that satisfies .
so . We compute
so the maximum length of is , and the answer is .
Suppose and intersect at and , respectively, and let and . Since is the midpoint of arc , bisects , and we get To find , we note that and , so
Writing , we can substitute known values and multiply the equations to get
The value we wish to maximize is
By the AM-GM inequality, , so giving the answer of . Equality is achieved when subject to the condition , which occurs for and .
Solution 5 (Projective)
By Pythagoras in we get
Since cross ratios are preserved upon projecting, note that By definition of a cross ratio, this becomes Let such that We know that so the LHS becomes
In the RHS, we are given every value except for However, Ptolemy's Theorem on gives Substituting, we get where we use
Again using we have Then Since this is a function in we differentiate WRT to find its maximum. By quotient rule, it suffices to solve Substituting back yields so is the answer.
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