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2018-05-28 Advanced

         

Ram has 2018 coins.

The 1st1^\text{st} coin has a 12\frac{1}{2} chance of flipping heads, the 2nd2^\text{nd} coin has a 13\frac{1}{3} chance of flipping heads, the 3rd3^\text{rd} coin has a 15\frac{1}{5} chance of flipping heads, continuing such that the kthk^\text{th} coin has a 1p(k)\frac{1}{p(k)} chance of flipping heads, where p(k)p(k) is the kthk^\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 m,1 \text{ m}, we choose a length uniformly at random between 0 m0 \text{ m} and 1 m,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 aa chosen with uniform distribution over the interval (0,1n],\big(0,\frac{1}{n}\big], where nNn\in\mathbb{N}, what's the expected value of the remainder 1moda1\bmod a for a particular n?n?

For m>1m>1, it can be proven that the integer sequence fm(n)=gcd(n+m,mn+1)f_m(n) = \gcd(n+m,mn+1) has a fundamental period Tm.T_m. In other words, nN, fm(n+Tm)=fm(n).\forall n \in \mathbb{N}, \space f_m(n+T_m) = f_m(n). Find an expression for TmT_m in terms of m,m, and then give your answer as T12.T_{12}.

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

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

Which is larger, f(2017)f(2017) or f(2018)?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 1Ω.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 Ω.\Omega.

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