HMMT 二月 2012 · 冲刺赛 · 第 7 题

HMMT February 2012 — Guts Round — Problem 7

[ 3 ] M is an 8 × 8 matrix. For 1 ≤ i ≤ 8, all entries in row i are at least i , and all entries on column i are at least i . What is the minimum possible sum of the entries of M ?

解析

[ 3 ] M is an 8 × 8 matrix. For 1 ≤ i ≤ 8, all entries in row i are at least i , and all entries on column i are at least i . What is the minimum possible sum of the entries of M ? Answer: 372 Let s be the minimum possible sum for an n by n matrix. Then, we note that n increasing it by adding row n + 1 and column n + 1 gives 2 n + 1 additional entries, each of which has 2 minimal size at least n + 1. Consequently, we obtain s = s + (2 n + 1)( n + 1) = s + 2 n + 3 n + 1. n +1 n n 2 2 Since s = 0 , we get that s = 2(7 + . . . + 0 ) + 3(7 + . . . + 0) + 8 = 372. 0 8