HMMT 二月 2006 · 冲刺赛 · 第 39 题

英文解析

39. A fat coin is one which, when tossed, has a 2 / 5 probability of being heads, 2 / 5 ofbeing tails, and 1 / 5 of landing on its edge. Mr. Fat starts at 0 on the real line. Everyminute, he tosses a fat coin. If it’s heads, he moves left, decreasing his coordinate by
    1; if it’s tails, he moves right, increasing his coordinate by 1. If the coin lands on itsedge, he moves back to 0. If Mr. Fat does this ad infinitum , what fraction of his timewill he spend at 0?
    Answer:1
    Solution: For n ∈ Z , let a be the fraction of the time Mr. Fat spends at n . By 3
    symmetry, a = a for all n .nn − n
    2 2 5
    For n > 0, we have a = a + a , or a = a − a . This Fibonacci-liken n − 1 n +1 n +1 n n − 1
    5 5 2
    recurrence can be solved explicitly to obtain
    | n | −| n |
    a = α · 2 + β · 2
    for all n ∈ Z . Now we also haven
    ∑
    a = 1 ,
    n ∈ Znβso we better have α = 0, so that a = β and a = . Now we also have a =
    0 ± 1 0
    ∑2
    2 2 1 1
    a + a + , so β = . This matches perfectly with a = 1 .
    − 1 1 nn ∈ Z
    5 5 5 3