HMMT 十一月 2012 · GEN 赛 · 第 8 题

英文解析

8. [ 7 ] Let n be the 200 th smallest positive real solution to the equation x − = tan x . Find the greatestn 2
   integer that does not exceed .
   π2
   Answer: 314 Drawing the graphs of the functions y = x − and y = tan x , we may observe that the
   ( )2
   (2 k − 1) π (2 k +1) πgraphs intersect exactly once in each of the intervals , for each k = 1 , 2 , · · · . Hence, the
   2 2
   399 π 401 π π
   200 th intersection has x in the range ( , ). At this intersection, y = x − is large, and thus, the
   2 2 2
   401 π 401 π π πintersection will be slightly less than . We have that ⌊ ⌋ = ⌊ 100 π + ⌋ = ⌊ 314 . 16+ ⌋ = 314.
   2 4 4 4