1971 AHSME Problems/Problem 17
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[hide]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 1
Note that, before adding the secant line, we have regions, and, starting from the edge of the disk, every time the secant line intersects a radius or the other end of the disk, a new region is created. Thus, we desire to maximize the number of radii that the secant line passes through.
The angle between any two adjacent radii must be . A given secant cannot pass through more than of the radii, and passing through exactly would imply that the secant must go through the center of the circle, and so it would not produce any new regions. Thus, to maximize the number of regions, the secant line must pass through radii (spanning degrees) and thereby create new regions (one for each radius and one for exiting the disk). Thus, we have a maximum of regions.
Solution 2
We can draw the cases for small values of 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 | |
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