Difference between revisions of "2017 AIME I Problems/Problem 6"
(Created page with "==Problem 6== A circle is circumscribed around an isosceles triangle whose two congruent angles have degree measure <math>x</math>. Two points are chosen independently and uni...") |
(No difference)
|
Revision as of 17:33, 8 March 2017
Problem 6
A circle is circumscribed around an isosceles triangle whose two congruent angles have degree measure . Two points are chosen independently and uniformly at random on the circle, and a chord is drawn between them. The probability that the chord intersects the triangle is . Find the difference between the largest and smallest possible values of .
Solution
The probability that the chord doesn't intersect the triangle is . The only way this can happen is if the two points are chosen on the same arc between two of the triangle vertices. The probability that a point is chosen on one of the arcs opposite one of the base angles is , and the probability that a point is chosen on the arc between the two base angles is . Therefore, we can write This simplifies to Which factors as So . The difference between these is .