HMMT 二月 2015 · 几何 · 第 5 题

英文解析

5. Let I be the set of points ( x, y ) in the Cartesian plane such that
   ( )
   1 / 4
   y 4
   x > + 2015
   2 2 29
   Let f ( r ) denote the area of the intersection of I and the disk x + y ≤ r of radius r > 0 centered at the origin (0 , 0). Determine the minimum possible real number L such that f ( r ) < Lr for all r > 0.2
   Answer: Let B ( P, r ) be the (closed) disc centered at P with radius r . Note that for allπ
   ( )3
   1 / 4
   √
   | y |4
   ′ ′y
   √
   ( x, y ) ∈ I , x > 0, and x > + 2015 > . Let I = { ( x, y ) : x 3 > | y |} . Then I ⊆ I and the
   ′ π 2 π 239
   intersection of I with B ((0 , 0) , r ) is r , so the f ( r ) = the area of I ∩ B ((0 , 0) , r ) is also less than r .
   3 3
   Thus L = works.π
   ( )3
   1 / 4
   ( ) ( )
   4 1 / 4
   | y | | y | | y |4
   4 y
   √ √
   On the other hand, if x > +7, then x > +7 > + 7 > + 2015 , which means
   9 9
   3 3
   √
   ′′ ′′ ′ ′′
   that if I = { ( x, y ) : ( x − 7) 3 > | y |} , then I ⊆ I . However, for r > 7, the area of I ∩ B ((7 , 0) , r − 7)
   π π
   2 ′′ 2
   is ( r − 7) , and I ⊆ I , B ((7 , 0) , r − 7) ⊆ B ((0 , 0) , r ), which means that f ( r ) > ( r − 7) for all
   3 3
   r > 7, from which it is not hard to see that L = is the minimum possible L .π
   √3
   4 4
   Remark: The lines y = ± 3 x are actually asymptotes for the graph of 9 x − y = 2015. The bulk of
   √
   α αthe problem generalizes to the curve | 3 x | − | y | = C (for a positive real α > 0 and any real C ); thecase α = 0 is the most familiar case of a hyperbola.
   √
   °