HMMT 十一月 2018 · 冲刺赛 · 第 2 题

HMMT November 2018 — Guts Round — Problem 2

[ 5 ] Pascal has a triangle. In the n th row, there are n + 1 numbers a , a , a , . . . , a where n, 0 n, 1 n, 2 n,n a = a = 1. For all 1 ≤ k ≤ n − 1, a = a − a . What is the sum of all numbers in n, 0 n,n n,k n − 1 ,k n − 1 ,k − 1 the 2018th row?

解析

[ 5 ] Pascal has a triangle. In the n th row, there are n + 1 numbers a , a , a , . . . , a where n, 0 n, 1 n, 2 n,n a = a = 1. For all 1  k  n 1, a = a a . What is the sum of all numbers in n, 0 n,n n,k n 1 ,k n 1 ,k 1 the 2018th row? Proposed by: Michael Ren Answer: 2 In general, the sum of the numbers on the n th row will be n n 1 X X a = a + ( a a ) + a = a + ( a a ) + a = 2 . n,k n, 0 n 1 ,k n 1 ,k 1 n,n n, 0 n 1 ,n 1 n 1 , 0 n,n k =0 k =1