# Problems of the Week

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You have 6 coins of distinct weights and a beam balance with two pans. You devise an algorithm that identifies the second lightest coin with complete certainty with at most $$M$$ weighings.

What is the minimum possible value of $$M?$$

Consider trying this problem first.

A line drawn through the centroid of a triangle intersects two sides internally, forming a yellow triangle and a green quadrilateral, as shown below. What is the minimum possible ratio of the yellow area to the green area?

A continuous function $$f(x)$$ is defined as follows for some non-zero constants $$a,$$ $$b,$$ and $$c:$$ f(x)=\cases{\begin{align} &x+a &&\text{for }|x|<2\\&bf\Big(\frac{x}{2}\Big)+c &&\text{for }|x|\ge2. \end{align}} Find the value of $$\frac{100}{a}+\frac{100}{b}+\frac{100}{c}.$$

On an infinitely large chess board, is there a configuration of kings and knights such that each king attacks exactly 2 kings and 2 knights, and each knight attacks exactly 2 kings and 2 knights?

This problem is part of the new Brilliant.org Open Problems Group (see Open Problem #1).

Call a non-empty set $$\mathbb V$$ of non-zero integers victorious if there exists a polynomial $$P(x)$$ with integer coefficients such that $$P(0) = 330$$ and $$P(v) = 2|v|$$ for all elements $$v \in \mathbb V.$$

How many victorious sets exist?

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