HMMT 二月 2009 · 代数 · 第 10 题
HMMT February 2009 — Algebra — Problem 10
题目详情
- [ 8 ] Let f ( x ) = 2 x − 2 x . For what positive values of a do there exist distinct b, c, d such that ( a, f ( a )), ( b, f ( b )), ( c, f ( c )), ( d, f ( d )) is a rectangle?
解析
- [ 8 ] Let f ( x ) = 2 x − 2 x . For what positive values of a do there exist distinct b, c, d such that ( a, f ( a )), ( b, f ( b )), ( c, f ( c )), ( d, f ( d )) is a rectangle? √ 3 Answer: [ , 1] 3 Solution: Say we have four points ( a, f ( a )), ( b, f ( b )), ( c, f ( c )), ( d, f ( d )) on the curve which form a rectangle. If we interpolate a cubic through these points, that cubic will be symmetric around the center of the rectangle. But the unique cubic through the four points is f ( x ), and f ( x ) has only one point of symmetry, the point (0 , 0). So every rectangle with all four points on f ( x ) is of the form ( a, f ( a )), ( b, f ( b )), ( − a, f ( − a )), ( − b, f ( − b )), and without loss of generality we let a, b > 0. Then for any choice of a and b these points form a paral- lelogram, which is a rectangle if and only if the distance from ( a, f ( a )) to (0 , 0) is equal to the distance 2 2 6 4 2 from ( b, f ( b )) to (0 , 0). Let g ( x ) = x + ( f ( x )) = 4 x − 8 x + 5 x , and consider g ( x ) restricted to x ≥ 0. We are looking for all the values of a such that g ( x ) = g ( a ) has solutions other than a . 2 3 2 Note that g ( x ) = h ( x ) where h ( x ) = 4 x − 8 x + 5 x . This polynomial h ( x ) has a relative maximum 1 5 of 1 at x = and a relative minimum of 25 / 27 at x = . Thus the polynomial h ( x ) − h (1 / 2) has 2 6 2 2 the double root 1 / 2 and factors as (4 x − 4 x + 1)( x − 1), the largest possible value of a for which 2 2 2 h ( x ) = h ( a ) is a = 1, or a = 1. The smallest such value is that which evaluates to 25 / 27 other than √ √ 2 3 3 5 / 6, which is similarly found to be a = 1 / 3, or a = . Thus, for a in the range ≤ a ≤ 1 the 3 3 equation g ( x ) = g ( a ) has nontrivial solutions and hence an inscribed rectangle exists. 3