HMMT 十一月 2011 · 冲刺赛 · 第 26 题

HMMT November 2011 — Guts Round — Problem 26

[ 12 ] Determine the positive real value of x for which √ √ √ 2 + AC + 2 Cx + AC − 2 + 2 Ax = 2( A + C ) x + 2 AC.

解析

[ 12 ] Determine the positive real value of x for which √ √ √ 2 + AC + 2 Cx + AC − 2 + 2 Ax = 2( A + C ) x + 2 AC. √ √ √ Answer: 4 Note that if we have a + b = a + b for non-negative reals a, b , then squaring gives √ us that 2 ab = 0, so that either a = 0 or b = 0. Now, note that (2 + AC + 2 Cx ) + ( AC − 2 + 2 Ax ) = (2( A + C ) x + 2 AC ) . Guts Round Consequently, either (2 + AC + 2 Cx ) or ( AC − 2 + 2 Ax ) must be equal to 0. However, we observe from the problems that both A , C , and x must be non-negative, so (2 + AC + 2 Cx ) > 0. As a result, we know that AC − 2 + 2 Ax = 0, or that 2 − AC B = x = . 2 A If we solve our system of equations for A, B, C , we get that B = 4.