HMMT 十一月 2012 · GEN 赛 · 第 8 题

HMMT November 2012 — GEN Round — Problem 8

[ 7 ] Let n be the 200th smallest positive real solution to the equation x − = tan x . Find the greatest 2 n integer that does not exceed . 2

解析

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