Difference between revisions of "2017 AIME I Problems/Problem 6"
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Revision as of 15:29, 5 June 2020
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
.
Note:
We actually do not need to spend time factoring . Since the problem asks for
, where
and
are the roots of the quadratic, we can utilize Vieta's by noting that
. Vieta's gives us
and
Plugging this into the above equation and simplifying gives us
or
.
Our answer is then .
See Also
2017 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 5 |
Followed by Problem 7 | |
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.