HMMT 十一月 2016 · 团队赛 · 第 9 题

英文解析

9. [ 7 ] A cylinder with radius 15 and height 16 is inscribed in a sphere. Three congruent smaller spheresof radius x are externally tangent to the base of the cylinder, externally tangent to each other, andinternally tangent to the large sphere. What is the value of x ?
   Proposed by: Eshaan Nichani
   √
   15 37 − 75
   Answer:
   Let O be the center of the large sphere, and let O , O , O be the centers of the small spheres. Consider 4
   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
   √22
   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
   √31
   2 24
   (8 + r ) + r . Now draw the array OO , and suppose it intersects the large sphere again at P .
   Then P is the point of tangency between the large sphere and the small sphere with center O , so 31
   √1
   √
   2 2 2 24
   OP = 15 + 8 = 17 = OO + O P = (8 + r ) + r + r . We rearrange this to be
   1 1
   √3
   2 24
   17 − r = (8 + r ) + r
   2 273
   ⇐⇒ 289 − 34 r + r = r + 16 r + 64
   ⇐⇒ r + 50 r − 225 = 0243
   √3 − 50 ± 50 + 4 · · 22524 = ⇒ r =3
   2 ·4
   √3
   15 37 − 75 = . 4