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24

Part of 2013 Online Math Open Problems

Problems(2)

2012-2013 Winter OMO #24

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1/16/2013
For a permutation π\pi of the integers from 1 to 10, define S(π)=∑i=19(π(i)−π(i+1))⋅(4+π(i)+π(i+1)), S(\pi) = \sum_{i=1}^{9} (\pi(i) - \pi(i+1))\cdot (4 + \pi(i) + \pi(i+1)), where π(i)\pi (i) denotes the iith element of the permutation. Suppose that MM is the maximum possible value of S(π)S(\pi) over all permutations π\pi of the integers from 1 to 10. Determine the number of permutations π\pi for which S(π)=MS(\pi) = M.
Ray Li
Online Math Open
2013-2014 Fall OMO #24

Source:

10/30/2013
The real numbers a0,a1,…,a2013a_0, a_1, \dots, a_{2013} and b0,b1,…,b2013b_0, b_1, \dots, b_{2013} satisfy an=1632n+2+an−1a_{n} = \frac{1}{63} \sqrt{2n+2} + a_{n-1} and bn=1962n+2−bn−1b_{n} = \frac{1}{96} \sqrt{2n+2} - b_{n-1} for every integer n=1,2,…,2013n = 1, 2, \dots, 2013. If a0=b2013a_0 = b_{2013} and b0=a2013b_0 = a_{2013}, compute ∑k=12013(akbk−1−ak−1bk). \sum_{k=1}^{2013} \left( a_kb_{k-1} - a_{k-1}b_k \right). Proposed by Evan Chen
Online Math OpenOmo