Difference between revisions of "1971 AHSME Problems/Problem 17"
Coolmath34 (talk | contribs) (Created page with "== Problem == A circular disk is divided by <math>2n</math> equally spaced radii(<math>n>0</math>) and one secant line. The maximum number of non-overlapping areas into whi...") |
m (see also, boxed answer) |
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It seems that for <math>2n</math> radii, there are <math>3n+1</math> distinct areas. The secant line must pass through <math>n</math> radii for this to occur. | It seems that for <math>2n</math> radii, there are <math>3n+1</math> distinct areas. The secant line must pass through <math>n</math> radii for this to occur. | ||
| − | The answer is <math>\textbf{(E)} | + | The answer is <math>\boxed{\textbf{(E) } 3n+1}.</math> |
-edited by coolmath34 | -edited by coolmath34 | ||
| + | |||
| + | == See Also == | ||
| + | {{AHSME 35p box|year=1971|num-b=16|num-a=18}} | ||
| + | {{MAA Notice}} | ||
Revision as of 08:04, 5 August 2024
Problem
A circular disk is divided by 
equally spaced radii(
) and one secant line. The maximum number of non-overlapping
areas into which the disk can be divided is
Solution
We can draw the cases for small values of 
![\[n = 0 \rightarrow \text{areas} = 1\]](http://latex.artofproblemsolving.com/1/b/4/1b400bb2c5a506970dea9b4f262de1533b485433.png)
![\[n = 1 \rightarrow \text{areas} = 4\]](http://latex.artofproblemsolving.com/0/c/4/0c4c477dd467136ec568600e06bc98e87353069f.png)
![\[n = 2 \rightarrow \text{areas} = 7\]](http://latex.artofproblemsolving.com/6/a/1/6a1082b2c4e1d929333f2dd11247d7d82142c9d2.png)
![\[n = 3 \rightarrow \text{areas} = 10\]](http://latex.artofproblemsolving.com/6/e/9/6e911b745e326ea1cf16f35a0aa4617c910c8143.png)
It seems that for radii, there are 
distinct areas. The secant line must pass through 
radii for this to occur.
The answer is 
-edited by coolmath34
See Also
| 1971 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 16 |
Followed by Problem 18 | |
| 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 • 31 • 32 • 33 • 34 • 35 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
