HMMT 二月 2013 · 代数 · 第 1 题

HMMT February 2013 — Algebra — Problem 1

Let x and y be real numbers with x > y such that x y + x + y + 2 xy = 40 and xy + x + y = 8. Find the value of x .

解析

Let x and y be real numbers with x > y such that x y + x + y + 2 xy = 40 and xy + x + y = 8. Find the value of x . √ 2 2 Answer: 3 + 7 We have ( xy ) + ( x + y ) = 40 and xy + ( x + y ) = 8. Squaring the second equation and subtracting the first gives xy ( x + y ) = 12 so xy, x + y are the roots of the quadratic 2 a − 8 a + 12 = 0. It follows that { xy, x + y } = { 2 , 6 } . If x + y = 2 and xy = 6, then x, y are the roots of 2 the quadratic b − 2 b + 6 = 0, which are non-real, so in fact x + y = 6 and xy = 2, and x, y are the roots √ √ 2 6+ 28 of the quadratic b − 6 b + 2 = 0. Because x > y , we take the larger root, which is = 3 + 7. 2