HMMT 十一月 2009 · 冲刺赛 · 第 9 题

英文解析

9. [ 7 ] Daniel wrote all the positive integers from 1 to n inclusive on a piece of paper. After carefulobservation, he realized that the sum of all the digits that he wrote was exactly 10,000. Find n .
   ∑x
   Answer: 799 Let S ( n ) denote the sum of the digits of n , and let f ( x ) = S ( n ). (We may addn =0
   n = 0 because S (0) = 0.) Observe that:
   ( )
   9 9 9 9
   ∑ ∑ ∑ ∑
   f (99) = ( a + b ) = 10 b + 10 a = 900
   a =0 a =0
   b =0 b =0
   If a is an integer between 1 and 9 inclusive, then:
   100 a +99 100 a +99
   ∑ ∑
   S ( n ) = ( a + S ( n − 100 a )) = 100 a + f (99) = 100 a + 900
   n =100 a n =100 a
   Summing, we get:
   ∑af (100 a + 99) = (100 a + 900) = 900( a + 1) + 50 a ( a + 1)
   n =0
   This formula can be used to find benchmarks. However, it turns out that this formula alone will suffice,
   as things turn out rather nicely:
   900( a + 1) + 50 a ( a + 1) = 10000
   50 a + 950 a + 900 = 100002
   50 a + 950 a − 9100 = 02
   50( a + 26)( a − 7) = 0
   a = 7
   Therefore f (799) = 10000, and our answer is 799.
   2 ANNUAL HARVARD-MIT NOVEMBER TOURNAMENT, 7 NOVEMBER 2009 — GUTS ROUNDnd