MathDB
2015-2016 Fall OMO #27

Source:

November 18, 2015
Online Math Open

Problem Statement

For integers 0≤m,n≤640 \le m,n \le 64, let α(m,n)\alpha(m,n) be the number of nonnegative integers kk for which ⌊m/2k⌋\left\lfloor m/2^k \right\rfloor and ⌊n/2k⌋\left\lfloor n/2^k \right\rfloor are both odd integers. Consider a 65×6565 \times 65 matrix MM whose (i,j)(i,j)th entry (for 1≤i,j≤651 \le i, j \le 65) is (−1)α(i−1,j−1). (-1)^{\alpha(i-1, j-1)}. Compute the remainder when det⁡M\det M is divided by 10001000.
Proposed by Evan Chen
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