Difference between revisions of "2012 AIME I Problems/Problem 13"
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==Problem 13== | ==Problem 13== | ||
− | Three concentric circles have radii <math>3,</math> <math>4,</math> and <math>5.</math> An equilateral triangle with one vertex on each circle has side length <math>s.</math> The largest possible area of the triangle can be written as <math>a + \ | + | Three concentric circles have radii <math>3,</math> <math>4,</math> and <math>5.</math> An equilateral triangle with one vertex on each circle has side length <math>s.</math> The largest possible area of the triangle can be written as <math>a + \tfrac{b}{c} \sqrt{d},</math> where <math>a,</math> <math>b,</math> <math>c,</math> and <math>d</math> are positive integers, <math>b</math> and <math>c</math> are relatively prime, and <math>d</math> is not divisible by the square of any prime. Find <math>a+b+c+d.</math> |
== Solution == | == Solution == |
Revision as of 22:59, 17 March 2012
Problem 13
Three concentric circles have radii and An equilateral triangle with one vertex on each circle has side length The largest possible area of the triangle can be written as where and are positive integers, and are relatively prime, and is not divisible by the square of any prime. Find
Solution
Reinterpret the problem in the following manner. Equilateral triangle has a point on the interior such that and A clockwise rotation about vertex maps to and to Note that angle is and which tells us that triangle is equilateral and that We now notice that and which tells us that angle is because there is a -- Pythagorean triple. Now note that and Applying the law of cosines on triangle yields
and thus the area of equals
so our final answer is
See also
2012 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |