HMMT 二月 2011 · ALGGEO 赛 · 第 1 题

英文解析

1. Determine the smallest value of x for which every distortion of H is necessarily convex.
   Answer: 4
   ′
   A A
   X X
   1 6
   A A
   2 5
   ′
   A A
   3 Y 4Y
   ′
   Let H = A A A A A A be the hexagon, and for all 1 ≤ i ≤ 6, let points A be considered such that
   1 2 3 4 5 6
   ′ ′ ′ ′ ′ ′ ′ ′i
   A A < 1. Let H = A A A A A A , and consider all indices modulo 6. For any point P in the plane,
   i 1 2 3 4 5 6 i
   ′
   let D ( P ) denote the unit disk { Q | P Q < 1 } centered at P ; it follows that A ∈ D ( A ).
   ′ ′ii
   Let X and X be points on line A A , and let Y and Y be points on line A A such that A X =
   1 6 3 4 1
   ′ ′ ′ ′
   A X = A Y = A Y = 1 and X and X lie on opposite sides of A and Y and Y lie on opposite sides
   1 3 3 1
   ′ ′ ′ ′
   of A . If X and Y lie on segments A A and A A , respectively, then segment A A lies between the
   3 1 6 3 4
   1 3
   ′ ′ xlines XY and X Y . Note that is the distance from A to A A .
   2 1 3
   Algebra & Geometry Individual Test 2
   ′
   A A
   X 1 X 6
   A A
   2 5
   ′
   Y A A
   3 Y 4
   If ≥ 2, then C ( A ) cannot intersect line XY , since the distance from XY to A A is 1 and thex
   2 1 3
   ′ ′ ′2
   distance from XY to A is at least 1. Therefore, A A separates A from the other 3 vertices of the
   1 3 22
   ′
   hexagon. By analogous reasoning applied to the other vertices, we may conclude that H is convex.
   ′ ′x
   If < 2, then C ( A ) intersects XY , so by choosing A = X and A = Y , we see that we may choose
   1 32
   ′ ′2
   A on the opposite side of XY , in which case H will be concave. Hence the answer is 4, as desired. 2