2022 SSMO Relay Round 5 Problems/Problem 3

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Problem

Let $T=$ TNYWR, and let $a_k=\text{cis}\left(\frac{k\pi}{T+1}\right)$. Suppose that\[\sum_{k=1}^{T+1} \frac{|a_{2k}+a_{2k+2}-a_{2k-1+T}|}{|a_{2k+1}-( a_{2k}+a_{2k+2})|}\]can be expressed in the form of $a+b\cos(\frac{\pi}{c})$, where $\text{cis}(x) = \cos(x) + i\sin(x)$. Find $a+b+c$.

Solution