HMMT 十一月 2016 · 冲刺赛 · 第 2 题

英文解析

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 8 th row and 8 th 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 + xi,j i +1 ,j +1 i +1 ,j i,j +1 i,jonly 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 holdsi +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,jthe 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