PUMaC 2015 · 团队赛 · 第 5 题

PUMaC 2015 — Team Round — Problem 5

[ 4 ] Imagine a regular a 2015-gon with edge length 2. At each vertex, draw a unit circle centered at that vertex and color the circle’s circumference orange. Now, another unit circle S is placed inside the polygon such that it is externally tangent to two adjacent circles centered at the vertices. This circle S is allowed to roll freely in the interior of the polygon as long as it remains externally tangent to the vertex circles. As it rolls, S turns the color of any point it touches into black. After it rolls completely around the interior of the polygon, the total length of the pπ black lengths can be expressed in the form for positive integers p, q satisfying gcd( p, q ) = 1. q What is p + q ? n

解析

[ 4 ] Imagine a regular a 2015-gon with edge length 2. At each vertex, draw a unit circle centered at that vertex and color the circle’s circumference orange. Now, another unit circle S is placed inside the polygon such that it is externally tangent to two adjacent circles centered at the vertices. This circle S is allowed to roll freely in the interior of the polygon as long as it remains externally tangent to the vertex circles. As it rolls, S turns the color of any point it touches into black. After it rolls completely around the interior of the polygon, the total length of the pπ black lengths can be expressed in the form for positive integers p, q satisfying gcd( p, q ) = 1. q What is p + q ? Solution: Figure 1: Three consecurtive circles in 2015-gon We determine how much the circumference of one of the unit circles is colored black and by symmetry, this is the same length for each unit circle so we just need to multiply that answer by 2015. If we look at figure 1, we see the length of the circumference colored black is the arc π length of the sector with central angle m ∠ CBE . We know that m ∠ ABC = m ∠ DBE = 3 2013 · π and m ∠ ABD = . So the total length for one circle and thus all 2015 circles is: 2015 [ ] 2013 · π 2 π 6039 π 4030 π 2009 π − · 2015 = − = 2015 3 3 3 3 So p + q = 2012 . Author: Roy Zhao n