HMMT 二月 2004 · 冲刺赛 · 第 32 题

英文解析

32. Define the sequence b , b , . . . , b by
    0 1 59
    {
    1 if i is a multiple of 3
    b =
    0 otherwise.i
    Let { a } be a sequence of elements of { 0 , 1 } such thatib ≡ 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 nb − a − a , these two numbers completely determine the solution { a } beginningn − 1 n − 2 n − 1 iwith 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 possibleivalue 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.