HMMT 二月 2008 · 代数 · 第 9 题

HMMT February 2008 — Algebra — Problem 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 .

解析

[ 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 . 1 2 Answer: After dividing the equation by x , we can rearrange it as 4 ( ) ( ) 2 1 1 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: 2 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 the two roots is √ 2 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. 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