HMMT 十一月 2013 · THM 赛 · 第 8 题

HMMT November 2013 — THM Round — Problem 8

[ 5 ] Let b ( n ) be the number of digits in the base − 4 representation of n . Evaluate b ( i ). i =1

解析

[ 5 ] Let b ( n ) be the number of digits in the base − 4 representation of n . Evaluate b ( i ). i =1 Answer: 12345 We have the following: • b ( n ) = 1 for n between 1 and 3. 2 2 2 • b ( n ) = 3 for n between 4 − 3 · 4 = 4 and 3 · 4 + 3 = 51. (Since a · 4 − b · 4 + c takes on 3 · 4 · 4 distinct values over 1 ≤ a ≤ 3, 0 ≤ b ≤ 3, 0 ≤ c ≤ 3, with minimum 4 and maximum 51.) 4 3 4 2 • b ( n ) = 5 for n between 4 − 3 · 4 − 3 · 4 = 52 and 3 · 4 + 3 · 4 + 3 = 819. 6 5 3 1 6 4 2 • b ( n ) = 7 for n between 4 − 3 · 4 − 3 · 4 − 3 · 4 = 820 and 3 · 4 + 3 · 4 + 3 · 4 + 3 > 2013. Thus 2013 ∑ b ( i ) = 7(2013) − 2(819 + 51 + 3) = 14091 − 2(873) = 14091 − 1746 = 12345 . i =1