HMMT 十一月 2016 · 团队赛 · 第 9 题

HMMT November 2016 — Team Round — Problem 9

[ 7 ] A cylinder with radius 15 and height 16 is inscribed in a sphere. Three congruent smaller spheres of radius x are externally tangent to the base of the cylinder, externally tangent to each other, and internally tangent to the large sphere. What is the value of x ?

解析

[ 7 ] A cylinder with radius 15 and height 16 is inscribed in a sphere. Three congruent smaller spheres of radius x are externally tangent to the base of the cylinder, externally tangent to each other, and internally tangent to the large sphere. What is the value of x ? Proposed by: Eshaan Nichani √ 15 37 − 75 Answer: 4 Let O be the center of the large sphere, and let O , O , O be the centers of the small spheres. Consider 1 2 3 G , the center of equilateral 4 O O O . Then if the radii of the small spheres are r , we have that 1 2 3 √ 2 r 2 2 √ OG = 8+ r and O O = O O = O O = 2 r , implying that O G = . Then OO = OG + OO = 1 2 2 3 3 1 1 1 1 3 √ 4 2 2 (8 + r ) + r . Now draw the array OO , and suppose it intersects the large sphere again at P . 1 3 Then P is the point of tangency between the large sphere and the small sphere with center O , so 1 √ √ 4 2 2 2 2 OP = 15 + 8 = 17 = OO + O P = (8 + r ) + r + r . We rearrange this to be 1 1 3 √ 4 2 2 17 − r = (8 + r ) + r 3 7 2 2 ⇐⇒ 289 − 34 r + r = r + 16 r + 64 3 4 2 ⇐⇒ r + 50 r − 225 = 0 3 √ 4 2 − 50 ± 50 + 4 · · 225 3 = ⇒ r = 4 2 · 3 √ 15 37 − 75 = . 4