HMMT 十一月 2012 · THM 赛 · 第 4 题
HMMT November 2012 — THM Round — Problem 4
题目详情
- [ 7 ] Find the sum of all real solutions for x to the equation ( x + 2 x + 3) = 2012.
解析
- [ 7 ] Find the sum of all real solutions for x to the equation ( x + 2 x + 3) = 2012. y 2 y Answer: − 2 When y = x + 2 x + 3, note that there is a unique real number y such that y = 2012 y y 2 because y is increasing in y . The sum of the real distinct solutions of the equation x + 2 x + 3 = y is 2 − 2 by Vieta’s Formulae as long as 2 + 4( y − 3) > 0, which is equivalent to y > 2. This is easily seen to be the case; therefore, our answer is − 2.