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Difference between revisions of "2022 SSMO Relay Round 5 Problems"
 
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==Problem 2==
 
==Problem 2==
  
−
Let <math>T=</math> TNYWR, and let <math>S=\{a_1,a_2,\dots,a_{2022}\}</math> be a sequence of 2022 positive integers such that <math>a_1\le a_2\le \cdots \le a_{2022}</math> and <math>\text{lcm}(a_1,a_2,\dots,a_{2022})=70T</math>. Also, <math>\text{gcd}(a_i,a_j)=1</math> for all <math>1\le i<j\le2022</math>. Find the number of possible sequences <math>S</math>.
+
Let <math>T=TNYWR</math>, and let <math>S=\{a_1,a_2,\dots,a_{2022}\}</math> be a sequence of 2022 positive integers such that <math>a_1\le a_2\le \cdots \le a_{2022}</math> and <math>\text{lcm}(a_1,a_2,\dots,a_{2022})=70T</math>. Also, <math>\text{gcd}(a_i,a_j)=1</math> for all <math>1\le i<j\le2022</math>. Find the number of possible sequences <math>S</math>.
  
 
[[2022 SSMO Relay Round 5 Problems/Problem 2|Solution]]
 
[[2022 SSMO Relay Round 5 Problems/Problem 2|Solution]]
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==Problem 3==
 
==Problem 3==
  
−
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>.  
  
 
[[2022 SSMO Relay Round 5 Problems/Problem 3|Solution]]
 
[[2022 SSMO Relay Round 5 Problems/Problem 3|Solution]]

Latest revision as of 19:18, 2 May 2025

Problem 1

Consider an $8\times 8$ chessboard with a knight in one of the center squares. The knight may move in an $L$-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?

Solution

Problem 2

Let $T=TNYWR$, and let $S=\{a_1,a_2,\dots,a_{2022}\}$ be a sequence of 2022 positive integers such that $a_1\le a_2\le \cdots \le a_{2022}$ and $\text{lcm}(a_1,a_2,\dots,a_{2022})=70T$. Also, $\text{gcd}(a_i,a_j)=1$ for all $1\le i<j\le2022$. Find the number of possible sequences $S$.

Solution

Problem 3

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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