HMMT 二月 2007 · COMB 赛 · 第 2 题

HMMT February 2007 — COMB Round — Problem 2

[ 3 ] How many 5-digit numbers abcde exist such that digits b and d are each the sum of the digits to their immediate left and right? (That is, b = a + c and d = c + e .)

解析

[ 3 ] How many 5-digit numbers abcde exist such that digits b and d are each the sum of the digits to their immediate left and right? (That is, b = a + c and d = c + e .) Answer: 330 . Note that a > 0, so that b > c , and e ≥ 0 so that d ≥ c . Conversely, for each choice of ( b, c, d ) with b > c and d ≥ c , there exists a unique pair ( a, e ) such that abcde is a number having the desired property. Thus, we compute 9 9 ∑ ∑ 2 (9 − c )(10 − c ) = c − 19 c + 90 = 330 . c =0 c =0