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Difference between revisions of "2022 SSMO Relay Round 5 Problems/Problem 3"

(Created page with "==Problem== Let <math>T=</math> TNYWR, and let <math>a_k=\text{cis}\left(\frac{k\pi}{T+1}\right)</math>. Suppose that<cmath>\sum_{k=1}^{T+1} \frac{|a_{2k}+a_{2k+2}-a_{2k-1+T}|...")
 
 
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==Problem==
 
==Problem==
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Let <math>T=</math> TNYWR, and let <math>a_k=\text{cis}\left(\frac{k\pi}{T+1}\right)</math>. Suppose that<cmath>\sum_{k=1}^{T+1} \frac{|a_{2k}+a_{2k+2}-a_{2k-1+T}|}{|a_{2k+1}-( a_{2k}+a_{2k+2})|}</cmath>can be expressed in the form of <math>a+b\cos(\frac{\pi}{c})</math>, where <math>\text{cis}(x) = \cos(x) + i\sin(x)</math>. Find <math>a+b+c</math>.
+
Let <math>T=TNYWR</math>, and let <math>a_k=\text{cis}\left(\frac{k\pi}{T+1}\right)</math>. Suppose that<cmath>\sum_{k=1}^{T+1} \frac{|a_{2k}+a_{2k+2}-a_{2k-1+T}|}{|a_{2k+1}-( a_{2k}+a_{2k+2})|}</cmath>can be expressed in the form of <math>a+b\cos(\frac{\pi}{c})</math>, where <math>\text{cis}(x) = \cos(x) + i\sin(x)</math>. Find <math>a+b+c</math>.
  
 
==Solution==
 
==Solution==

Latest revision as of 19:24, 2 May 2025

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

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