Difference between revisions of "1996 AHSME Problems/Problem 22"
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Revision as of 05:55, 16 April 2012
Problem
Four distinct points, 
, 
, 
, and 
, are to be selected from 
points
evenly spaced around a circle. All quadruples are equally likely to be chosen.
What is the probability that the chord 
intersects the chord 
?
Solution
Let 
be a convex cyclic quadrilateral inscribed in a circle. There are 
ways to divide the points into two groups of two.
If you pick 
and 
, you have two sides of the quadrilateral, which do not intersect.
If you pick 
and 
, you have the other two sides of the quadrilateral, which do not intersect.
If you pick 
and 
, you have the diagonals of the quadrilateral, which do intersect.
Any four points on the original circle of can be connected to form such a convex quarilateral , and only placing and as one of the diagonals of the figure will form intersecting chords. Thus, the answer is 
, which is option 
.
Notice that is irrelevant to the solution of the problem; in fact, you may pick points from the entire circumference of the circle.
See also
| 1996 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 21 |
Followed by Problem 23 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
| All AHSME Problems and Solutions | ||
