HMMT 二月 2020 · 冲刺赛 · 第 12 题

HMMT February 2020 — Guts Round — Problem 12

[6] An 11 × 11 grid is labeled with consecutive rows 0 , 1 , 2 , . . . , 10 and columns 0 , 1 , 2 , . . . , 10 so that it is 10 filled with integers from 1 to 2 , inclusive, and the sum of all of the numbers in row n and in column n n are both divisible by 2 . Find the number of possible distinct grids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT February 2020, February 15, 2020 — GUTS ROUND Organization Team Team ID# √

解析

[6] An 11 × 11 grid is labeled with consecutive rows 0 , 1 , 2 , . . . , 10 and columns 0 , 1 , 2 , . . . , 10 so that 10 it is filled with integers from 1 to 2 , inclusive, and the sum of all of the numbers in row n and in n column n are both divisible by 2 . Find the number of possible distinct grids. Proposed by: Joseph Heerens 1100 Answer: 2 Solution: We begin by filling the 10 by 10 grid formed by rows and columns 1 through 10 with any 10 100 1000 values, which we can do in (2 ) = 2 ways. Then in column 0, there is at most 1 way to fill in the 10 square in row 10, 2 ways for the square in row 9, down to 2 ways in row 0. Similarly, there is 1 way to fill in the square in row 0 and column 10, 2 ways to fill in the square in row 0 and column 9, etc. Overall, 1 2 3 9 10 9 8 1 100 the number of ways to fill out the squares in row or column 0 is 2 · 2 · 2 · · · 2 · 2 · 2 · 2 · · · 2 = 2 , 1000 100 1100 so the number of possible distinct grids 2 · 2 = 2 . √