HMMT 十一月 2014 · 冲刺赛 · 第 11 题

HMMT November 2014 — Guts Round — Problem 11

[ 8 ] How many integers n in the set { 4 , 9 , 14 , 19 , . . . , 2014 } have the property that the sum of the decimal digits of n is even?

解析

[ 8 ] How many integers n in the set { 4 , 9 , 14 , 19 , . . . , 2014 } have the property that the sum of the decimal digits of n is even? Answer: 201 We know that 2014 does not qualify the property. So, we’ll consider { 4 , 9 , 14 , ..., 2009 } instead. Now, we partition this set into 2 sets: { 4 , 14 , 24 , ..., 2004 } and { 9 , 19 , 29 , ..., 2009 } . For each so the first and second set are basically x 4 and x 9, where x = 0 , 1 , 2 , ..., 200, respectively. And we know that for each value of x , x must be either even or odd, which makes exactly one of { x 4 , x 9 } has even sum of decimal digits. Therefore, there are in total of 201 such numbers.