HMMT 二月 2001 · 冲刺赛 · 第 22 题

英文解析

22. [6] A man is standing on a platform and sees his train move such that after tseconds it is 2 t + d feet from his original position, where d is some number. Call the 2
    0 0
    smallest (constant) speed at which the man have to run so that he catches the train v . Interms of n , find the n th smallest value of d that makes v a perfect square.
    Solution: The train’s distance from the man’s original position is t + d , and the man’s 20
    distance from his original position if he runs at speed v is vt at time t . We need to find 0
    where t + d = vt has a solution. Note that this is a quadratic equation with discriminant 2
    √0
    √
    D = v − 4 d , so it has solutions for real D , i.e. where v ≥ 4 d , so 4 d must be a perfect 2
    0 0 0
    square. This happens when 4 d is an even power of 2: the smallest value is 2 , the second 0
    2 4 2( n − 1) n − 10
    smallest is 2 , the third smallest is 2 , and in general the n th smallest is 2 , or 4 .