PUMaC 2008 · 几何（B 组） · 第 8 题

PUMaC 2008 — Geometry (Division B) — Problem 8

(5 points) Infinitesimal Randall Munroe is glued to the center of a pentagon with side length 1. At each corner of the pentagon is a confused infinitesimal velociraptor. At any time, each raptor is running at one unit per second directly towards the next raptor in the pentagon (in counterclockwise order). How far does each confused raptor travel before it reaches Randall Munroe?

解析

How many ordered pairs of real numbers ( x, y ) are there such that x + y = 200 and √ √ 2 2 2 2 ( x − 5) + ( y − 5) + ( x + 5) + ( y + 5) is an integer? ( ANS: 12 √ √ 2 2 2 2 The locus of points such that L = ( x − 5) + ( y − 5) + ( x + 5) + ( y + 5) is an ellipse with 2 2 foci at (5 , 5) and ( − 5 , − 5), and the locus of points such that x + y = 200 is a circle of radius √ √ 10 2 centered at the origin. The smallest L such that the ellipse will intersect the circle is 20 2, √ and the largest is 10 10. For L strictly between these values, the ellipse will intersect the circle √ √ exactly 4 times. There are three integers between 20 2 and 10 10, so there are 4 · 3 = 12 points √ √ 2 2 2 2 on the circle such that ( x − 5) + ( y − 5) + ( x + 5) + ( y + 5) is an integer. CB: GL, ACH) 3 Geometry