HMMT 二月 2010 · TEAM1 赛 · 第 6 题

英文解析

6. [ 20 ] Let S be a convex set in the plane with a finite area a . Prove that either a = 0 or S is bounded.
   Note: a set is bounded if it is contained in a circle of finite radius. Note: a set is convex if, whenevertwo points A and B are in the set, the line segment between them is also in the set.
   Solution: If all points in S lie on a straight line, then a = 0.
   Otherwise we may pick three points A , B , and C that are not collinear. Let ω be the incircle of 4 ABC ,
   with I its center and r its radius. Since S is convex, S must contain ω .
   ωYAX
   BZIM
   √Ca 2
   Suppose S also contains a point X at a distance d from I , with d > R . We will show that d ≤ r + ,2
   r 2
   which implies that the S is bounded since all points are contained within the circle centered at I of
   √
   a 2
   radius r + .2
   r 2
   Let Y and Z be on ω such that XY and XZ are tangents to ω . Because S is convex, it must containkite IY XZ , whose area we can compute in terms of d and r .
   IM IY
   Let M be the midpoint of Y Z . Since 4 IY X ∼ 4 IM Y , we know that = , that is, IM =
   IY IX
   √
   2 √
   ( IY )42
   r r r 1
   2 2 = . Then M Y = r − = r 1 − ( ) = Y Z .
   IX d d d 22
   √
   √
   1 r
   2 2 2
   The area of IY XZ is ( Y Z )( IX ) = rd 1 − ( ) = r d − r . This must be less than or equal to a ,
   2 d
   √
   2 2 2 4 2 2 a a 22
   the area of S . This yields a ≥ r d − r or d ≤ r + . It follows that d ≤ r + , as desired.2
   2 2
   r r