HMMT 二月 2008 · 冲刺赛 · 第 19 题

HMMT February 2008 — Guts Round — Problem 19

[ 10 ] Let ABCD be a regular tetrahedron, and let O be the centroid of triangle BCD . Consider the point P on AO such that P minimizes P A + 2( P B + P C + P D ). Find sin ∠ P BO .

解析

[ 10 ] Let ABCD be a regular tetrahedron, and let O be the centroid of triangle BCD . Consider the point P on AO such that P minimizes P A + 2( P B + P C + P D ). Find sin ∠ P BO . 1 Answer: We translate the problem into one about 2-D geometry. Consider the right triangle 6 ABO , and P is some point on AO . Then, the choice of P minimizes P A + 6 P B . Construct the line 1 ` through A but outside the triangle ABO so that sin ∠ ( AO, ` ) = . For whichever P chosen, let 6 1 Q be the projection of P onto , then P Q = AP . Then, since P A + 6 P B = 6( P Q + P B ), it is 6 equivalent to minimize P Q + P B . Observe that this sum is minimized when B, P, Q are collinear and the line through them is perpendicular to ` (so that P Q + P B is simply the distance from B to ` ). ◦ ◦ Then, ∠ AQB = 90 , and since ∠ AOB = 90 as well, we see that A, Q, P, B are concyclic. Therefore, 1 ∠ P BO = ∠ OP A = ∠ ( AO, ), and the sine of this angle is therefore . 6