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2018-04-16 Advanced

         

An equilateral triangular plate has side length aa, and the moment of inertia about the axis shown is II.

A regular hexagonal plate has side length 2a2a, and the moment of inertia about the axis shown can be given as a constant multiple of II: kIk I.

What is k?k?

Note: Assume both plates have negligible thickness.

Let aa be a positive integer such that the following limit exists: limx1(1x11xa2x+1).\lim_{x\to 1} \left( \frac{1}{x-1} - \frac{1}{x^a-2x+1}\right). If the value of the limit is b,b, find a+b.a+b.

Concatenating four copies of 23 produces 23232323=23232323.23 || 23 || 23 || 23 = 23232323.

Now, suppose you concatenate xx copies of any positive integer n.n.

What is the minimum value of xx such that the result of this concatenation is guaranteed to be 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)T(n) be the number of such nn-digit sequences. What is limnT(n+1)T(n)? \lim_{n \to \infty} \frac{T(n+1)}{T(n)}?

There exist positive integers A,B,C,D,E,FA,B,C,D,E,F such that p=1ncot6(pπ2n+1)=1Fn(2n1)(An4+Bn3+Cn2Dn+E) \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 nn, where FF is as small as possible.

What is A+B+C+D+E+F?A + B + C + D + E + F?


Bonus: Use this result to prove that p=11p6=1945π6. \sum_{p=1}^\infty \tfrac{1}{p^6} = \tfrac{1}{945}\pi^6.

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