PUMaC 2008 · 几何（B 组） · 第 7 题

PUMaC 2008 — Geometry (Division B) — Problem 7

(5 points) 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?

解析

(4 points) Circles A , B , and C each have radius r , and their centers are the vertices of an equilateral triangle of side length 6 r . Two lines are drawn, one tangent to A and C and one tangent to B and C , such that A is on the opposite side of each line from B and C . Find the sine of the angle between the two lines. 2 Geometry A C B 1 √ 2 6 − 1 ( ANS: . 6 The line tangent to B and C is parallel to the line between the centers of B and C . The line tangent to A and C passes through the midpoint between the centers of A and C , hence the angle − 1 r it makes with the line between their centers is sin . Hence we seek 3 r √ π − 1 r π − 1 1 π − 1 1 − 1 2 sin( − sin = sin( ) cos( − sin ) + cos( ) sin( − sin . Using cos(sin ( x )) = 1 − x 3 3 r 3 3 3 3 √ √ √ 3 1 1 1 2 6 − 1 2 gives 1 − ( ) − = , as desired. CB: IAF, ACH) 2 3 2 3 6 2 2