PUMaC 2016 · 团队赛 · 第 2 题

PUMaC 2016 — Team Round — Problem 2

(3) Temerant is a spherical planet with radius 1000 kilometers. The government wants to build twelve towers of the same height on the equator of Temerant, so that every point on the equator can be seen from at least one tower. The minimum possible height of the towers can √ √ be written, in kilometers, as a b − c d − e for positive integers a , b , c , d , and e (with b and d not divisible by the square of any prime). Compute a + b + c + d + e .

解析

Each tower needs to see an arc of measure 30 , i.e. an 15 arc on either side, so that every 1000 ◦ point is covered twice. This means that = cos 15 , where h is the height of the towers. 1000+ h ( ) √ √ √ √ ◦ ◦ ◦ 2 3 1 6+ 2 h Note that cos 15 = cos(45 − 30 ) = + = , and so we have 1 + = 2 2 2 4 1000 √ √ √ √ 1000+ h 4 √ √ = = 6 − 2. Thus, h = 1000 6 − 1000 2 − 1000, and so our answer is 1000 6+ 2 1000 + 6 + 1000 + 2 + 1000 = 3008 . Problem written by Eric Neyman.