Difference between revisions of "1996 AIME Problems/Problem 15"
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The only value of <math>\theta</math> that fits in this context comes from <math>4 \cos^2 2\theta - 3 = 0 \Longrightarrow \cos 2\theta = \frac{\sqrt{3}}{2} \Longrightarrow \theta = 15^{\circ}</math>. The answer is <math>\lfloor 1000r \rfloor = \left\lfloor 1000 \cdot \frac{180 - 5\theta}{180 - 3\theta} \right\rfloor = \left \lfloor \frac{7000}{9} \right \rfloor = \boxed{777}</math>. | The only value of <math>\theta</math> that fits in this context comes from <math>4 \cos^2 2\theta - 3 = 0 \Longrightarrow \cos 2\theta = \frac{\sqrt{3}}{2} \Longrightarrow \theta = 15^{\circ}</math>. The answer is <math>\lfloor 1000r \rfloor = \left\lfloor 1000 \cdot \frac{180 - 5\theta}{180 - 3\theta} \right\rfloor = \left \lfloor \frac{7000}{9} \right \rfloor = \boxed{777}</math>. | ||
− | === Solution 2 ( | + | === Solution 2 (trigonometry) === |
Define <math>\theta</math> as above. Since <math>\angle CAB = \angle CBO</math>, it follows that <math>\triangle COB \sim \triangle CBA</math>, and so <math>\frac{CO}{BC} = \frac{BC}{AC} \Longrightarrow BC^2 = AC \cdot CO = 2CO^2 \Longrightarrow BC = CO\sqrt{2}</math>. The [[Law of Sines]] on <math>\triangle BOC</math> yields that | Define <math>\theta</math> as above. Since <math>\angle CAB = \angle CBO</math>, it follows that <math>\triangle COB \sim \triangle CBA</math>, and so <math>\frac{CO}{BC} = \frac{BC}{AC} \Longrightarrow BC^2 = AC \cdot CO = 2CO^2 \Longrightarrow BC = CO\sqrt{2}</math>. The [[Law of Sines]] on <math>\triangle BOC</math> yields that | ||
Revision as of 20:51, 22 December 2015
Problem
In parallelogram , let be the intersection of diagonals and . Angles and are each twice as large as angle , and angle is times as large as angle . Find the greatest integer that does not exceed .
Contents
Solution
Solution 1 (trignometry)
Let . Then , , and . Since is a parallelogram, it follows that . By the Law of Sines on ,
Dividing the two equalities yields
Pythagorean and product-to-sum identities yield
and the double and triple angle () formulas further simplify this to
The only value of that fits in this context comes from . The answer is .
Solution 2 (trigonometry)
Define as above. Since , it follows that , and so . The Law of Sines on yields that
Expanding using the sine double and triple angle formulas, we have
By the quadratic formula, we have , so (as the other roots are too large to make sense in context). The answer follows as above.
Solution 3
We will focus on . Let , so . Draw the perpendicular from intersecting at . Without loss of generality, let . Then , since is the circumcenter of . Then .
By the Exterior Angle Theorem, and . That implies that . That makes . Then since by AA ( and reflexive on ), .
Then by the Pythagorean Theorem, . That makes equilateral. Then . The answer follows as above.
See also
1996 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Final Problem | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.