HMMT 十一月 2018 · 冲刺赛 · 第 28 题

HMMT November 2018 — Guts Round — Problem 28

[ 15 ] What is the 3-digit number formed by the 9998 through 10000 digits after the decimal point 1 in the decimal expansion of ? 998 Note: Make sure your answer has exactly three digits, so please include any leading zeroes if necessary.

解析

[ 15 ] What is the 3-digit number formed by the 9998 through 10000 digits after the decimal point 1 in the decimal expansion of ? 998 Note: Make sure your answer has exactly three digits, so please include any leading zeroes if necessary. Proposed by: Sujay Kazi Answer: 042 1 1 250 1 Note that + = repeats every 498 digits because 499 is prime, so does as well (after the 998 2 499 998 th th first 498 block). Now we need to find 38 to 40 digits. We expand this as a geometric series 1 1 2 1000 = = . 001 + . 001 ⇥ . 002 + . 001 ⇥ . 002 + · · · . 2 998 1 1000 th th th nd st The contribution to the 36 through 39 digits is 4096, the 39 through 42 digits is 8192, and 41 th through 45 digits is 16384. We add these together: 4 0 9 6 8 1 9 2 1 6 8 3 4 4 1 0 4 2 0 · · · th The remaining terms decrease too fast to have e ↵ ect on the digits we are looking at, so the 38 to th 40 digits are 042.