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2017-11-20 Advanced

         

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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