Difference between revisions of "2009 AIME II Problems/Problem 10"
Aimesolver (talk | contribs) (New page: Let O be the intersection of BC and AD. By the Angle Bisector Theorem,5/BO = 13/CO, so BO = 5x and CO = 13x, and BO + OC = BC = 12, so x = 2/3, and OC = 26/3. Let P be the altitude fro...) |
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| − | + | Four lighthouses are located at points A, B, C, and D. The lighthouse A is 5 km away from the lighthouse at B, the lighthouse at B is 12 km from the lighthouse at C, and the lighthouse at A is 13 km from the lighthouse at C. To an observer at A, the angle determined by the lights at B and D and the angle determined by the lights at C and D are equal. To an observer at C, the angle determined by the lights at A and B are equal, and the angle determined by the lights at D and B are equal. The number of km from A to D is given by (p*sqrt (q))/r where p, q, and r are relatively prime positive integers, and r is not divisible by the square of any prime. Find p+q+r. | |
Revision as of 18:35, 17 April 2009
Four lighthouses are located at points A, B, C, and D. The lighthouse A is 5 km away from the lighthouse at B, the lighthouse at B is 12 km from the lighthouse at C, and the lighthouse at A is 13 km from the lighthouse at C. To an observer at A, the angle determined by the lights at B and D and the angle determined by the lights at C and D are equal. To an observer at C, the angle determined by the lights at A and B are equal, and the angle determined by the lights at D and B are equal. The number of km from A to D is given by (p*sqrt (q))/r where p, q, and r are relatively prime positive integers, and r is not divisible by the square of any prime. Find p+q+r.
