2017 AIME I Problems/Problem 6
Contents
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 .
Solution 2 (Not Complementary Counting method)
Because we know that we have an isosceles triangle with angles of (and we know that x is an inscribed angle), that means that the arc that is intercepted by this angle is
. We form this same conclusion for the other angle
, and
. Therefore we get
arcs, namely,
,
, and
. To have the chords intersect the triangle, we need the two points selected (to make a chord) to be on completely different arcs. An important idea to understand is that order matters in this case, so we have the equation
*
*
+
*
*
*
=
which using trivial algebra gives you
and factoring gives you
and so your answer is
.
~jske25
Video Solution
https://youtu.be/Mk-MCeVjSGc?t=690 ~Shreyas S
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 |
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