HMMT 二月 2023 · 冲刺赛 · 第 16 题

HMMT February 2023 — Guts Round — Problem 16

[14] The graph of the equation x + y = b x + y c consists of several line segments. Compute the sum of their lengths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT February 2023, February 18, 2023 — GUTS ROUND Organization Team Team ID#

解析

[14] The graph of the equation x + y = b x + y c consists of several line segments. Compute the sum of their lengths. Proposed by: Sean Li √ √ Answer: 4 + 6 − 2 2 2 2 2 Solution: We split into cases on the integer k = b x + y c . Note that x + y = k but x + y ≥ 1 2 1 2 2 2 ( x + y ) = k and x + y < k + 1, which forces k ≤ 2. 2 2 √ √ 2 2 2 2 If k = 0, the region defined by 0 ≤ x + y < 1 and x + y = 0 is the diameter from ( , − ) to 2 2 √ √ 2 2 ( − , ), which has length 2. 2 2 2 2 If k = 1, the region 1 ≤ x + y < 2 and x + y = 1 consists of two segments, which is the chord on √ √ √ √ 2 2 2 2 2 2 2 x + y = 2 minus the chord on x + y = 1. The former has length 2 ( 2) − ( ) = 6, and the 2 √ √ √ √ √ 2 2 2 latter has length 2 1 − ( ) = 2. So the total length here is 6 − 2. 2 2 2 2 2 If k = 2, the region 2 ≤ x + y < 3 and x + y = 1 is the chord on x + y = 3, which has length √ √ √ 2 2 2 ( 3) − ( 2) = 2. √ √ √ √ Our final answer is 2 + ( 6 − 2) + 2 = 4 + 6 − 2.