Difference between revisions of "1996 AHSME Problems/Problem 22"
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| + | ==Problem== | ||
| + | Four distinct points, <math>A</math>, <math>B</math>, <math>C</math>, and <math>D</math>, are to be selected from <math>1996</math> points | ||
| + | evenly spaced around a [[circle]]. All quadruples are equally likely to be chosen. | ||
| + | What is the [[probability]] that the [[chord]] <math>AB</math> intersects the chord <math>CD</math>? | ||
| + | |||
| + | <math> \text{(A)}\ \frac{1}{4}\qquad\text{(B)}\ \frac{1}{3}\qquad\text{(C)}\ \frac{1}{2}\qquad\text{(D)}\ \frac{2}{3}\qquad\text{(E)}\ \frac{3}{4} </math> | ||
| + | |||
| + | ==Solution== | ||
| + | |||
| + | Let <math>WXYZ</math> be a convex [[cyclic quadrilateral]] inscribed in a circle. There are <math>\frac{\binom{4}{2}}{2} = 3</math> ways to divide the points into two groups of two. | ||
| + | |||
| + | If you pick <math>WX</math> and <math>YZ</math>, you have two sides of the quadrilateral, which do not intersect. | ||
| + | |||
| + | If you pick <math>XY</math> and <math>ZW</math>, you have the other two sides of the quadrilateral, which do not 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 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. | ||
| + | |||
==See also== | ==See also== | ||
{{AHSME box|year=1996|num-b=21|num-a=23}} | {{AHSME box|year=1996|num-b=21|num-a=23}} | ||
| + | |||
| + | [[Category:Introductory Geometry Problems]] | ||
| + | [[Category:Introductory Probability Problems]] | ||
| + | [[Category:Introductory Combinatorics Problems]] | ||
| + | {{MAA Notice}} | ||
Latest revision as of 20:43, 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. 
