2018-04-16 Advanced

         

An equilateral triangular plate has side length \(a\), and the moment of inertia about the axis shown is \(I\).

A regular hexagonal plate has side length \(2a\), and the moment of inertia about the axis shown can be given as a constant multiple of \(I\): \(k I\).

What is \(k?\)

Note: Assume both plates have negligible thickness.

Let \(a\) be a positive integer such that the following limit exists: \[\lim_{x\to 1} \left( \frac{1}{x-1} - \frac{1}{x^a-2x+1}\right).\] If the value of the limit is \(b,\) find \(a+b.\)

Concatenating four copies of 23 produces \(23 || 23 || 23 || 23 = 23232323.\)

Positive integer \(n\) is of unknown digit length.

What is the minimum number of copies of \(n\) that you need to concatenate to ensure the result is a multiple of 11?

Find the number of 10-digit sequences where the difference between any two consecutive digits is 1, using only the digits 1, 2, 3, and 4.

Examples of such 10-digit sequences are 1234321232 and 2121212121.


Bonus: Let \(T(n)\) be the number of such \(n\)-digit sequences. What is \( \lim_{n \to \infty} \frac{T(n+1)}{T(n)}?\)

There exist positive integers \(A,B,C,D,E,F\) such that \[ \sum_{p=1}^n \cot^6 \left(\tfrac{p \pi}{2n+1}\right) = \tfrac{1}{F}n(2n-1)\big(An^4 + Bn^3 + Cn^2 - Dn + E\big) \] for all positive integers \(n\), where \(F\) is as small as possible.

What is \(A + B + C + D + E + F?\)


Bonus: Use this result to prove that \[ \sum_{p=1}^\infty \tfrac{1}{p^6} = \tfrac{1}{945}\pi^6. \]

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