MathDB
2017-2018 Spring OMO Problem 30

Source:

April 3, 2018

Problem Statement

Let p=2017p = 2017. Given a positive integer nn, an n×nn\times n matrix AA is formed with each element aija_{ij} randomly selected, with equal probability, from {0,1,,p1}\{0,1,\ldots,p - 1\}. Let qnq_n be probability that detA1(modp)\det A\equiv 1\pmod{p}. Let q=limnqnq=\displaystyle\lim_{n\rightarrow\infty} q_n. If d1,d2,d3,d_1, d_2, d_3, \ldots are the digits after the decimal point in the base pp expansion of qq, then compute the remainder when k=1p2dk\displaystyle\sum_{k = 1}^{p^2} d_k is divided by 10910^9.
Proposed by Ashwin Sah