HMMT 二月 2004 · 冲刺赛 · 第 32 题

HMMT February 2004 — Guts Round — Problem 32

[10] Define the sequence b , b , . . . , b by 0 1 59 { 1 if i is a multiple of 3 b = i 0 otherwise. Let { a } be a sequence of elements of { 0 , 1 } such that i b ≡ a + a + a (mod 2) n n − 1 n n +1 for 0 ≤ n ≤ 59 ( a = a and a = a ). Find all possible values of 4 a + 2 a + a . 0 60 − 1 59 0 1 2

解析

Define the sequence b , b , . . . , b by 0 1 59 { 1 if i is a multiple of 3 b = i 0 otherwise. Let { a } be a sequence of elements of { 0 , 1 } such that i b ≡ a + a + a (mod 2) n n − 1 n n +1 for 0 ≤ n ≤ 59 ( a = a and a = a ). Find all possible values of 4 a + 2 a + a . 0 60 − 1 59 0 1 2 Solution: 0 , 3 , 5 , 6 Try the four possible combinations of values for a and a . Since we can write a ≡ 0 1 n b − a − a , these two numbers completely determine the solution { a } beginning n − 1 n − 2 n − 1 i with them (if there is one). For a = a = 0, we can check that the sequence beginning 0 1 0 , 0 , 0 , 0 , 1 , 1 and repeating every 6 indices is a possible solution for { a } , so one possible i value for 4 a + 2 a + a is 0. The other three combinations for a and a similarly lead 0 1 2 0 1 to valid sequences (produced by repeating the sextuples 0 , 1 , 1 , 1 , 0 , 1; 1 , 0 , 1 , 1 , 1 , 0; 1 , 1 , 0 , 1 , 0 , 1, respectively); we thus obtain the values 3, 5, and 6.