MathDB

Let

\Lambda= \{1, 2, \ldots, 2v-1,2v\}

and

P=\{\alpha_1, \alpha_2, \ldots, \alpha_{2v-1}, \alpha_{2v}\}

be a permutation of the elements of

\Lambda

.
(a) Prove that

\sum_{i=1}^v \alpha_{2i-1}\alpha_{2i} \leq \sum_{i=1}^v (2i-1)2i.

(b) Determine the largest positive integer

m

such that we can partition the

m\times m

square into

7

rectangles for which every pair of them has no common interior points and their lengths and widths form the following sequence:

1,2,3,4,5,6,7,8,9,10,11,12,13,14.

Problem Statement