Difference between revisions of "2007 UNCO Math Contest II Problems/Problem 8"
(Created page with "== Problem == A regular decagon <math>P_1P_2P_3\cdots P_{10}</math> is drawn in the coordinate plane with <math>P_1</math> at <math>(2,0)</math> and <math>P_6</math> at <math>(8...") |
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== Solution == | == Solution == | ||
| + | <math>9,706,576</math> | ||
| + | Translate the center of the decagon to the origin. Now the vertices represent the roots | ||
| + | of <math>f(x)=x^{10}-3^{10}=0</math>. Since the <math>P_n</math> are each <math>5</math> more than the roots of <math>f(x) = 0</math> , they would be | ||
| + | the roots of<math>f(x-5)=0</math> or <math>(x-5)^{10}-3^{10}=0</math>. The product then is the constant term, or | ||
| + | <math>5^{10}-3^{10}= 9,706,576</math> | ||
== See Also == | == See Also == | ||
Latest revision as of 04:32, 12 January 2019
Problem
A regular decagon 
is drawn
in the coordinate plane with 
at 
and 
at 
. If 
denotes the point
, compute the numerical value of
the following product of complex numbers:
where 
as usual.
![[asy] draw(polygon(10),dot); draw((-2,0)--(2,0),black); draw((-5/3,-2)--(-5/3,2),black); MP("P_1",(-1,0),NW);MP("(2,0)",(-.9,0),SW); MP("P_6",(1,0),NE);MP("(8,0)",(.9,0),SE); [/asy]](http://latex.artofproblemsolving.com/a/c/d/acd58d961876928535be1ba8fa362bc000ac8b1c.png)
Solution
Translate the center of the decagon to the origin. Now the vertices represent the roots
of 
. Since the are each 
more than the roots of 
, they would be
the roots of
or 
. The product then is the constant term, or
See Also
| 2007 UNCO Math Contest II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 7 |
Followed by Problem 9 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 | ||
| All UNCO Math Contest Problems and Solutions | ||
