Difference between revisions of "2022 SSMO Relay Round 5 Problems"
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==Problem 3== | ==Problem 3== | ||
| − | Let <math>T=</math> TNYWR, and let <math>a_k=\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>\cis(x) = \cos(x) + i\sin(x)</math>. Find <math>a+b+c</math>. | + | 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>. |
[[2022 SSMO Relay Round 5 Problems/Problem 3|Solution]] | [[2022 SSMO Relay Round 5 Problems/Problem 3|Solution]] | ||
Latest revision as of 20:59, 31 May 2023
Problem 1
Consider an 
chessboard with a knight in one of the center squares. The knight may move in an 
-shaped fashion, going two squares in one direction and one square in a perpendicular direction, but cannot go outside the chessboard. How many squares can the knight reach in exactly two moves?
Problem 2
Let 
TNYWR, and let 
be a sequence of 2022 positive integers such that 
and 
. Also, 
for all 
. Find the number of possible sequences 
.
Problem 3
Let TNYWR, and let 
. Suppose that ![\[\sum_{k=1}^{T+1} \frac{|a_{2k}+a_{2k+2}-a_{2k-1+T}|}{|a_{2k+1}-( a_{2k}+a_{2k+2})|}\]](http://latex.artofproblemsolving.com/a/f/e/afe05527b41754cd280412f4ab1c1dd39869a5c6.png)
can be expressed in the form of 
, where 
. Find 
.
