HMMT 十一月 2016 · 冲刺赛 · 第 2 题

HMMT November 2016 — Guts Round — Problem 2

The entry in the i th row and j th column equals the sum of the numbers in the ( i − 1) by ( j − 1) sub-grid with row less than i and column less than j . What is the number in the 8th row and 8th column? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT November 2016, November 12, 2016 — GUTS ROUND Organization Team Team ID#

解析

The entry in the i th row and j th column equals the sum of the numbers in the ( i − 1) by ( j − 1) sub-grid with row less than i and column less than j . What is the number in the 8th row and 8th column? Proposed by: Eshaan Nichani Answer: 2508 Let x be the number in the i th row and the j th column. Then if i, j ≥ 2, x − x − x + x i,j i +1 ,j +1 i +1 ,j i,j +1 i,j only counts the term x since every other term is added and subtracted the same number of times. i,j Thus x = x + x when i, j ≥ 2. Also, x = x = i so x = x + x holds i +1 ,j +1 i +1 ,j i,j +1 2 ,i i, 2 i +1 ,j +1 i +1 ,j i,j +1 for all i, j ≥ 1 except for ( i, j ) = (1 , 1) where x is one less than expected. This means that x is 2 , 2 i,j the number ways of travelling from (1 , 1) to ( i, j ), minus the number of ways of travelling from (2 , 2) ( ) ( ) 14 12 to ( i, j ), which is − = 2508 . 7 6