2022 SSMO Relay Round 5 Problems

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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2022 SSMO Relay Round 5 Problems

Problem 1

Problem 2

Problem 3