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2018-05-28 Advanced

         

Ram has 2018 coins.

The \(1^\text{st}\) coin has a \(\frac{1}{2}\) chance of flipping heads, the \(2^\text{nd}\) coin has a \(\frac{1}{3}\) chance of flipping heads, the \(3^\text{rd}\) coin has a \(\frac{1}{5}\) chance of flipping heads, continuing such that the \(k^\text{th}\) coin has a \(\frac{1}{p(k)}\) chance of flipping heads, where \(p(k)\) is the \(k^\text{th}\) prime number.

Ram flips each of his coins once.

What is the probability that Ram flips an even number of heads?

From a string of length \(1 \text{ m},\) we choose a length uniformly at random between \(0 \text{ m}\) and \(1 \text{ m},\) and cut as many of these lengths as possible from the string.

What is the expected length of the remaining string (in meters)?


Bonus: If we have a random number \(a\) chosen with uniform distribution over the interval \(\big(0,\frac{1}{n}\big],\) where \(n\in\mathbb{N}\), what's the expected value of the remainder \(1\bmod a\) for a particular \(n?\)

For \(m>1\), it can be proven that the integer sequence \(f_m(n) = \gcd(n+m,mn+1)\) has a fundamental period \(T_m.\) In other words, \[\forall n \in \mathbb{N}, \space f_m(n+T_m) = f_m(n).\] Find an expression for \(T_m\) in terms of \(m,\) and then give your answer as \(T_{12}.\)

\(f(n)\) is the number of intersections of diagonals inside a regular \(n\)-sided polygon.

For example, in the diagram, \(f(6)=13.\)

Which is larger, \(f(2017)\) or \(f(2018)?\)

The Brilliant logo is similar in appearance to a Disdyakis triacontahedron, a Catalan solid. Suppose we had a circuit consisting of the edges and vertices of one of these solids, and the resistance along any edge is \(1\, \Omega.\)

The North and South pole vertices of this solid each have 10 connected edges. Determine the equivalent resistance between the poles of the solid in \(\Omega.\)

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