HMMT 二月 2001 · 冲刺赛 · 第 22 题

HMMT February 2001 — Guts Round — Problem 22

[6] A man is standing on a platform and sees his train move such that after t 2 seconds it is 2 t + d feet from his original position, where d is some number. Call the 0 0 smallest (constant) speed at which the man have to run so that he catches the train v . In terms of n , find the n th smallest value of d that makes v a perfect square. 0

解析

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