2022 AMC 12B Problem 24

Question

The figure below depicts a regular

7

-gon inscribed in a unit circle.

What is the sum of the

4

th powers of the lengths of all

21

of its edges and diagonals?

49

98

147

168

196

A chord between vertices

d

apart has squared length

2 - 2\cos\dfrac{2\pi d}{7};

there are

7

of each type

d = 1, 2, 3

Use

\sum_{d=1}^{3} \cos\dfrac{2\pi d}{7} = -\dfrac12

to evaluate

\sum \left(2 - 2\cos\dfrac{2\pi d}{7}\right)^2

A chord joining two vertices

d

steps apart has squared length

2 - 2\cos\dfrac{2\pi d}{7},

and there are

7

chords for each of

d = 1, 2, 3.

The required sum is

7 \sum_{d=1}^{3} \left(2 - 2\cos\tfrac{2\pi d}{7}\right)^2.

Using

\displaystyle\sum_{d=1}^{3} \cos\tfrac{2\pi d}{7} = -\tfrac12

and

\displaystyle\sum_{d=1}^{3} \cos^2\tfrac{2\pi d}{7} = \tfrac54,

the inner sum expands to

4\left(3 + 1 + \tfrac54\right) = 21.

Therefore the total is

7 \cdot 21 = 147.

Thus, the correct answer is C.

Problem 24 in other years