HMMT 二月 2008 · 代数 · 第 9 题

英文解析

9. [ 7 ] Let S be the set of points ( a, b ) with 0 ≤ a, b ≤ 1 such that the equation
   4 3 2
   x + ax − bx + ax + 1 = 0
   has at least one real root. Determine the area of the graph of S .
   Answer: After dividing the equation by x , we can rearrange it as 21
   ( ) ( )4
   1 12
   x + + a x + − b − 2 = 0
   x x
   1 1
   Let y = x + . We can check that the range of x + as x varies over the nonzero reals is ( −∞ , − 2] ∪ [2 , ∞ ).
   x x
   Thus, the following equation needs to have a real root:
   y + ay − b − 2 = 0 .2
   Its discriminant, a + 4( b + 2), is always positive since a, b ≥ 0. Then, the maximum absolute value of 2
   the two roots is
   √
   a + a + 4( b + 2)2
   .
   We need this value to be at least 2. This is equivalent to 2
   √
   a + 4( b + 2) ≥ 4 − a.2
   We can square both sides and simplify to obtain
   2 a ≥ 2 − b
   This equation defines the region inside [0 , 1] × [0 , 1] that is occupied by S , from which we deduce that the desired area is 1 / 4. 2