Difference between revisions of "2014 AIME II Problems/Problem 14"

(Created page with "14. In △ABC, AB=10, ∠A=30∘, and ∠C=45∘. Let H, D, and M be points on the line <math><math>\overline{BC}</math></math> such that <math><math>\overline{AH}⊥</math>\over...")
 
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14. In △ABC, AB=10, ∠A=30∘, and ∠C=45∘. Let H, D, and M be points on the line <math><math>\overline{BC}</math></math> such that <math><math>\overline{AH}⊥</math>\overline{BC}<math></math>, <math></math>∠BAD=∠CAD<math></math>, and <math></math>BM=CM<math></math>. Point <math></math>N<math></math> is the midpoint of the segment <math></math>\overline{HM}<math></math>, and point <math></math>P<math></math> is on ray AD such that <math></math>\overline{PN}⊥\overline{BC}<math></math>. Then <math></math>AP^2=\dfrac{m}{n}$</math>, where m and n are relatively prime positive integers. Find m+n.
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14. In △ABC, AB=10, ∠A=30∘, and ∠C=45∘. Let H, D, and M be points on the line BC¯¯¯¯¯ such that AH¯¯¯¯¯¯⊥BC¯¯¯¯¯, ∠BAD=∠CAD, and BM=CM. Point N is the midpoint of the segment HM¯¯¯¯¯¯¯, and point P is on ray AD such that PN¯¯¯¯¯¯⊥BC¯¯¯¯¯. Then AP2=mn, where m and n are relatively prime positive integers. Find m+n.
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DPatrick 9:17:57 pm

Revision as of 22:18, 29 March 2014

14. In △ABC, AB=10, ∠A=30∘, and ∠C=45∘. Let H, D, and M be points on the line BC¯¯¯¯¯ such that AH¯¯¯¯¯¯⊥BC¯¯¯¯¯, ∠BAD=∠CAD, and BM=CM. Point N is the midpoint of the segment HM¯¯¯¯¯¯¯, and point P is on ray AD such that PN¯¯¯¯¯¯⊥BC¯¯¯¯¯. Then AP2=mn, where m and n are relatively prime positive integers. Find m+n. DPatrick 9:17:57 pm

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