HMMT 十一月 2020 · 冲刺赛 · 第 5 题

HMMT November 2020 — Guts Round — Problem 5

[6] The points (0 , 0) , (1 , 2) , (2 , 1) , (2 , 2) in the plane are colored red while the points (1 , 0) , (2 , 0) , (0 , 1) , (0 , 2) are colored blue. Four segments are drawn such that each one connects a red point to a blue point and each colored point is the endpoint of some segment. The smallest possible sum of the lengths of the √ segments can be expressed as a + b , where a, b are positive integers. Compute 100 a + b . 2 x z z

解析

[6] The points (0 , 0) , (1 , 2) , (2 , 1) , (2 , 2) in the plane are colored red while the points (1 , 0) , (2 , 0) , (0 , 1) , (0 , 2) are colored blue. Four segments are drawn such that each one connects a red point to a blue point and each colored point is the endpoint of some segment. The smallest possible sum of the lengths of the √ segments can be expressed as a + b , where a, b are positive integers. Compute 100 a + b . Proposed by: Andrew Gu Answer: 305 Solution: If (2 , 2) is connected to (0 , 1) or (1 , 0), then the other 6 points can be connected with segments of total √ length 3, which is minimal. This leads to a total length of 3 + 5. On the other hand, if (2 , 2) is connected to (0 , 2) or (0 , 2), then connecting the other points with seg- √ √ √ ments of total length 2 is impossible, so the minimal length is at least 2 + 2 + 2 = 4 + 2 > 3 + 5. 2 x z z