HMMT 十一月 2008 · 冲刺赛 · 第 23 题

HMMT November 2008 — Guts Round — Problem 23

[ 12 ] If x and y are real numbers such that + = 1, find the largest possible value of + . 4 9 4 9 1 k +1 k 1 2008

解析

[ 12 ] If x and y are real numbers such that + = 1, find the largest possible value of + . 4 9 4 9 Answer: 9 The first equation is an ellipse with major axis parallel to the y-axis. If the second expression is set equal to a certain value c , then it is also the equation of an ellipse with major axis parallel to the y-axis; further, it is similar to the first ellipse. So the largest value of c occurs when both ellipse are tangent on the x-axis, at x = 6 , y = 0, which gives 9 as the largest value of c . 1 k +1 k 1 2008