Difference between revisions of "1996 AHSME Problems/Problem 22"
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If you pick <math>WY</math> and <math>XZ</math>, you have the diagonals of the quadrilateral, which do intersect. | If you pick <math>WY</math> and <math>XZ</math>, you have the diagonals of the quadrilateral, which do intersect. | ||
− | Any four points on the original circle of <math>1996</math> can be connected to form such a convex | + | Any four points on the original circle of <math>1996</math> can be connected to form such a convex quadrilateral <math>WXYZ</math>, and only placing <math>A</math> and <math>C</math> as one of the diagonals of the figure will form intersecting chords. Thus, the answer is <math>\frac{1}{3}</math>, which is option <math>\boxed{\text{B}}</math>. |
Notice that <math>1996</math> is irrelevant to the solution of the problem; in fact, you may pick points from the entire [[circumference]] of the circle. | Notice that <math>1996</math> is irrelevant to the solution of the problem; in fact, you may pick points from the entire [[circumference]] of the circle. | ||
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[[Category:Introductory Geometry Problems]] | [[Category:Introductory Geometry Problems]] | ||
+ | [[Category:Introductory Probability Problems]] | ||
[[Category:Introductory Combinatorics Problems]] | [[Category:Introductory Combinatorics Problems]] | ||
− | + | {{MAA Notice}} |
Latest revision as of 21:43, 11 April 2019
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 quadrilateral
, 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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.