2015 AIME II Problems/Problem 11
Contents
Problem
The circumcircle of acute has center . The line passing through point perpendicular to intersects lines and and and , respectively. Also , , , and , where and are relatively prime positive integers. Find .
Diagram
Solution 1
Call and the feet of the altitudes from to and , respectively. Let . Notice that because both are right triangles, and . By , . However, since is the circumcenter of triangle , is a perpendicular bisector by the definition of a circumcenter. Hence, . Since we know and , we have . Thus, . .
Solution 2
Notice that , so . From this we get that . So , plugging in the given values we get , so , and .
Solution 3
Let . Drawing perpendiculars, and . From there, . Thus, . Using , we get . Now let's find . After some calculations with ~ , . Therefore, . .
Solution 4
Let . Extend to touch the circumcircle at a point . Then, note that . But since is a diameter, , implying . It follows that is a cyclic quadrilateral.
Let . By Power of a Point, The answer is .
Solution 5
Note: This is not a very good solution, but it is relatively natural and requires next to no thinking.
Denote the circumradius of to be , the circumcircle of to be , and the shortest distance from to circle to be .
Using Power of a Point on relative to circle , we get that . Using Pythagorean Theorem on triangle to get . Subtracting the first equation from the second, we get that and therefore . Now, set . Using law of cosines on to find in terms of and plugging that into the extended law of sines, we get . Squaring both sides and cross multiplying, we get . Now, we get using quadratic formula. If you drew a decent diagram, is acute and therefore (You can also try plugging in both in the end and seeing which gives a rational solution). Note that Using the cosine addition formula and then plugging in what we know about , we get that . Now, the hard part is to find what is. We therefore want . For the numerator, by inspection will not work for integers and . The other case is if there is . By inspection, works. Therefore, plugging all this in yields the answer, . Solution by hyxue
Solution 6
Reflect , across to points and , respectively with on the circle and collinear. Now, by parallel lines. From here, as collinear. From here, is cyclic, and by power of a point we obtain . ~awang11's sol
See also
2015 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 10 |
Followed by Problem 12 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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