Difference between revisions of "1981 USAMO Problems/Problem 1"
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== Solution == | == Solution == | ||
{{solution}} | {{solution}} | ||
| + | Let <math>n=3k+1</math>. Multiply throughout by <math>\pi/3n</math>. We get | ||
| + | <math>\frac{\pi}{3}</math>=<math>\frac{\pi \times k}{n}</math>+<math>\frac{\pi}{3n}</math> | ||
| + | |||
| + | Re-arranging, we get | ||
| + | |||
| + | <math>\frac{\pi}{3}</math>-<math>\frac{\pi \times k}{n}</math>=<math>\frac{\pi}{3n}</math> | ||
| + | |||
| + | A way to interpret it is that if we know the value <math>k</math>, then the reminder angle of subtracting <math>k</math> times the given angle from <math>\frac{\pi}{3}</math> gives us <math>\frac{\pi}{3n}</math>, the desired trisected angle. | ||
| + | |||
| + | This can be extended to the case when <math>n=3k+2</math> where now, the equation becomes | ||
| + | <math>\frac{\pi}{3}</math>-<math>\frac{\pi \times k}{n}</math>=<math>\frac{2\pi}{3n}</math> | ||
| + | |||
| + | Hence in this case, we will have to subtract <math>k</math> times the original angle from <math>\frac{\pi}{3}</math> to get twice the the trisected angle. We can bisect it after that to get the trisected angle. | ||
== See Also == | == See Also == | ||
{{USAMO box|year=1981|before=First Question|num-a=2}} | {{USAMO box|year=1981|before=First Question|num-a=2}} | ||
Revision as of 16:05, 4 June 2014
Problem
Prove that if 
is not a multiple of 
, then the angle 
can be trisected with ruler and compasses.
Solution
This problem needs a solution. If you have a solution for it, please help us out by adding it.
Let 
. Multiply throughout by 
. We get
=
+
Re-arranging, we get
-=
A way to interpret it is that if we know the value 
, then the reminder angle of subtracting times the given angle from gives us , the desired trisected angle.
This can be extended to the case when 
where now, the equation becomes
-=
Hence in this case, we will have to subtract times the original angle from to get twice the the trisected angle. We can bisect it after that to get the trisected angle.
See Also
| 1981 USAMO (Problems • Resources) | ||
| Preceded by First Question |
Followed by Problem 2 | |
| 1 • 2 • 3 • 4 • 5 | ||
| All USAMO Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
