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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: kI.
What is k?
Note: Assume both plates have negligible thickness.
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Let a be a positive integer such that the following limit exists: x→1lim(x−11−xa−2x+11). If the value of the limit is b, find a+b.
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Concatenating four copies of 23 produces 23∣∣23∣∣23∣∣23=23232323.
Now, suppose you concatenate x copies of any positive integer n.
What is the minimum value of x such that the result of this concatenation is guaranteed to be a multiple of 11?
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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 limn→∞T(n)T(n+1)?
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There exist positive integers A,B,C,D,E,F such that p=1∑ncot6(2n+1pπ)=F1n(2n−1)(An4+Bn3+Cn2−Dn+E) 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 p=1∑∞p61=9451π6.
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