HMMT 十一月 2024 · 冲刺赛 · 第 28 题

HMMT November 2024 — Guts Round — Problem 28

[15] The graph of the equation tan( x + y ) = tan( x ) + 2 tan( y ) , with its pointwise holes filled in, partitions the coordinate plane into congruent regions. Compute the perimeter of one of these regions. ◦

解析

[15] The graph of the equation tan( x + y ) = tan( x ) + 2 tan( y ), with its pointwise holes filled in, partitions the coordinate plane into congruent regions. Compute the perimeter of one of these regions. Proposed by: Karthik Venkata Vedula √ Answer: π ( 5 + 1) Solution: We manipulate the given equation as follows: tan( x + y ) = tan x + 2 tan y tan x + tan y = tan x + 2 tan y 1 − tan x tan y tan x + tan y = tan x + 2 tan y − tan x tan y tan x + 2 tan y tan x tan y tan x + 2 tan y = tan y tan x tan y tan( x + y ) = tan y tan y (tan x tan( x + y ) − 1) = 0 Thus, the graph of tan( x + y ) = tan x + 2 tan y is the union of • the graph of tan y = 0, which is equivalent to y = nπ for some n ∈ Z ; and π • the graph of tan( x + y ) = cot x , which is equivalent to 2 x + y = + nπ for some n ∈ Z . 2 2 π π − 2 π − π π 2 π 0 − π − 2 π Each of the above graphs is a disjoint union of equally spaced parallel lines. Thus, the entire graph partitions the plane into congruent parallelograms. To compute the perimeter, we need to pick two adjacent lines from each bullet point. We pick y = 0, y = π , and 2 x + y = ± π/ 2. This is a parallelogram with vertices ( ± π/ 4 , 0), ( − 3 π/ 4 , π ), √ √ and ( − π/ 4 , π ). This is a parallelogram with side lengths π/ 2 and π 5 / 2, so the perimeter is π ( 5+1). ◦