HMMT 二月 2015 · 冲刺赛 · 第 33 题

HMMT February 2015 — Guts Round — Problem 33

[ 25 ] Let N denote the sum of the decimal digits of . Estimate the value of N . If your answer is 100 a positive integer A written fully in decimal notation (for example, 521495223), your score will be the | A − N | greatest integer not exceeding 25 · (0 . 99) . Otherwise, your score will be zero. 2 2

解析

[ 25 ] Let N denote the sum of the decimal digits of . Estimate the value of N . If your answer is 100 a positive integer A written fully in decimal notation (for example, 521495223), your score will be the | A − N | greatest integer not exceeding 25 · (0 . 99) . Otherwise, your score will be zero. Answer: 621 http://www.wolframalpha.com/input/?i=sum+of+digits+of+nCr(1000,100) To see this, one can estimate there are about 150 digits, and we expect the digits to be roughly random, for 150 · 4 . 5 ≈ 675, which is already very close to the actual answer. The actual number of digits is 140, and here 140 · 4 . 5 = 630 is within 9 of the actual answer. Guts 2 2