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AMC (AIMO validation set) / https://artofproblemsolving.com/wiki/index.php/2022_AMC_12B_Problems/Problem_24 / The figure below depicts a regular 7-gon inscribed in a unit circle.
Problem
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problem
The figure below depicts a regular -gon inscribed in a unit circle.
[asy] import geometry; unitsize(3cm); draw(circle((0,0),1),linewidth(1.5)); for (int i = 0; i < 7; ++i) { for (int j = 0; j < i; ++j) { draw(dir(i * 360/7) -- dir(j * 360/7),linewidth(1.5)); } } for(int i = 0; i < 7; ++i) { dot(dir(i * 360/7),5+black); } [/asy]
What is the sum of the th powers of the lengths of all of its edges and diagonals?Output the remainder when the answer is divided by 100.
Plain-text mathematical notation (without MathML)
The figure below depicts a regular 7-gon inscribed in a unit circle.
[asy] import geometry; unitsize(3cm); draw(circle((0,0),1),linewidth(1.5)); for (int i = 0; i < 7; ++i) { for (int j = 0; j < i; ++j) { draw(dir(i * 360/7) -- dir(j * 360/7),linewidth(1.5)); } } for(int i = 0; i < 7; ++i) { dot(dir(i * 360/7),5+black); } [/asy]
What is the sum of the 4th powers of the lengths of all 21 of its edges and diagonals?Output the remainder when the answer is divided by 100.
Original LaTeX notation
The figure below depicts a regular $7$-gon inscribed in a unit circle.
[asy] import geometry; unitsize(3cm); draw(circle((0,0),1),linewidth(1.5)); for (int i = 0; i < 7; ++i) { for (int j = 0; j < i; ++j) { draw(dir(i * 360/7) -- dir(j * 360/7),linewidth(1.5)); } } for(int i = 0; i < 7; ++i) { dot(dir(i * 360/7),5+black); } [/asy]
What is the sum of the $4$th powers of the lengths of all $21$ of its edges and diagonals?Output the remainder when the answer is divided by 100.
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