Difference between revisions of "2022 AMC 12B Problems/Problem 24"
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==Video Solution== | ==Video Solution== | ||
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+ | https://youtu.be/nO5p_xfXykI | ||
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+ | ~ ThePuzzlr | ||
https://youtu.be/yRbweIYtLU8 | https://youtu.be/yRbweIYtLU8 | ||
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com) | ~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com) |
Revision as of 21:21, 17 November 2022
Contents
[hide]Problem
The figure below depicts a regular 7-gon inscribed in a unit circle.
What is the sum of the 4th powers of the lengths of all 21 of its edges and diagonals?
Solution (Complex numbers approach)
There are 7 segments whose lengths are , 7 segments whose lengths are , 7 segments whose lengths are .
Therefore, the sum of the 4th powers of these lengths is where the fourth from the last equality follows from the property that
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Solution (Trig approach)
There are 7 segments whose lengths are , 7 segments whose lengths are , 7 segments whose lengths are .
Therefore, the sum of the 4th powers of these lengths is where the second from the last equality follows from the property that
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Video Solution
~ ThePuzzlr
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)