HMMT 二月 2010 · 冲刺赛 · 第 17 题

英文解析

17. [ 9 ] An ant starts at the origin, facing in the positive x -direction. Each second, it moves 1 unit forward,
    − 1
    then turns counterclockwise by sin ( ) degrees. What is the least upper bound on the distance 3
    between the ant and the origin? (The least upper bound is the smallest real number r that is at least 5
    as big as every distance that the ant ever is from the origin.)
    √
    √
    Answer: 10 We claim that the points the ant visits lie on a circle of radius . We show this 10
    √2
    10 1 3
    by saying that the ant stays a constant distance from the point ( , ).
    2 2 2
    ′
    Suppose the ant moves on a plane P . Consider a transformation of the plane P such that after the first
    ′ ′ ′
    move, the ant is at the origin of P and facing in the direction of the x axis (on P ). The transformation
    ′
    to get from P to P can be gotten by rotating P about the origin counterclockwise through an angle − 1 3 1 3
    sin ( ) and then translating it 1 unit to the right. Observe that the point ( , ) is fixed under this
    5 2 2
    1 3 4 3 1 3
    transformation, which can be shown through the expression ( + i )( + i ) + 1 = + i . It follows
    2 2 5 5 2 2
    1 3
    that at every point the ant stops, it will always be the same distance from ( , ). Since it starts at
    2 2
    √
    (0 , 0), this fixed distance is .10
    Guts Round 2
    1 3
    ( , )
    2 2
    ′
    − 1 3 xx
    Since sin ( ) is not a rational multiple of π , the points the ant stops at form a dense subset of the circle in question. As a result, the least upper bound on the distance between the ant and the origin 5
    √
    is the diameter of the circle, which is 10. 1