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2018-09-10 Advanced


Given a cube with edge length ll and mass M,M, what is the moment of inertia about the space diagonal?

f(x)=exp(k=1x1k)f(x) = \exp \left(\sum_{k=1}^{\lceil{x}\rceil} \frac 1k \right) The graph of the above is drawn in red, where exp(x)=ex.\exp(x) = e^x. The purple line is a linear function of the form y=mx.y=mx. The value of mm is maximized such that the purple line never intersects one of the red segments.

What is 1000m?\lfloor 1000m \rfloor ?

The Fibonacci sequence can be defined with induction as Fn=Fn1+Fn2F_n=F_{n-1}+F_{n-2}, where F1=0F_1=0 and F2=1F_2=1. It is also well-known that the limit of the ratio between two consecutive terms limFn+1Fn\lim \frac{F_{n+1}}{F_n} is the golden ratio ϕ\phi.

In the generalized Fibonacci sequence below, the original Fibonacci sequence is Fn2.F_n^2. Fnm=i=1mFnim  with  Fnm=0  if  n<m  and  Fmm=1.F_n^m=\sum_{i=1}^{m}{F_{n-i}^m}\ \text{ with }\ F_n^m=0\ \text{ if } \ n<m\ \text{ and }\ F_m^m=1. Every sequence will also have a generalized golden ratio ϕm=limnFn+1mFnm.\phi_m=\displaystyle \lim_{n \rightarrow \infty}{ \frac{F_{n+1}^m}{F_{n}^m}}.

What is the limit of the generalized golden ratio below? limmϕm=limmlimnFn+1mFnm\lim_{m \rightarrow \infty}{\phi_m} = \lim_{m \rightarrow \infty}{\lim_{n \rightarrow \infty}{\frac{F_{n+1}^m}{F_{n}^m}}}

Three 66-sided fair dice are rolled and their sum is recorded. If the three dice show all different values, then stop. Otherwise, roll any dice with the same value again, and add the sum of these re-rolled dice to the previous sum. Continue this process until all three dice show different values, and let XX be the final total of the values of all the dice rolled.

Here is an example:

  • Roll 1: (5,5,3)(5,5,3) are thrown, for a running total of 13.13.
  • Roll 2: The two dice showing 55 are rolled again and come up (3,1),(3,1), for a running total of 13+4=17.13+4=17. The three dice now show (3,1,3).(3,1,3).
  • Roll 3: The two dice showing 33 are rolled again and come up (1,1),(1,1), for a running total of 17+2=19.17+2=19. The three dice now show (1,1,1).(1,1,1).
  • Roll 4: All three dice are rolled again and come up (5,5,6),(5,5,6), for a running total of 19+16=35.19+16=35.
  • Roll 5: The two dice showing 55 are rolled again and come up (1,5),(1,5), for a running total of 35+6=41.35+6=41. The three dice now show (1,5,6),(1,5,6), so we stop with X=41.X=41.

The expected value of XX can be written as ab,\frac{a}{b}, where aa and bb are coprime positive integers.

What is a+b?a+b?

This problem was inspired by a previous featured problem. In this version, all of the dice must show different numbers (not just the numbers that were just rolled).

A point is picked uniformly at random from the surface of an nn-dimensional unit hypercube centered at the origin.

What is the minimum nn for which the expected distance from the point to the origin is greater than 1?


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