Mock AIME 1 2007-2008 Problems/Problem 14
Problem 14
Points and lie on , with radius , so that is acute. Extend to point so that . Let be the intersection of and such that and . If can be written as , where and are relatively prime and is not divisible by the square of any prime, find .
Solution
EDIT: The following solution is wrong. See the discussion page.
By the cosine double-angle formula,
The Law of Cosines on with respect to yields
\begin{align*}r^2 &= r^2 + AB^2 - 2 \cdot AB \cdot r \cos \angle BAO \\ AB^2 &= 2 \cdot AB \cdot r \cdot \frac{\sqrt{11}}{4}\\ AB &= \frac{r\sqrt{11}}{2} (Error compiling LaTeX. )
Now, . The Law of Cosines on with respect to yields The answer is thus .
See also
Mock AIME 1 2007-2008 (Problems, Source) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |