PUMaC 2010 · 代数（A 组） · 第 3 题

PUMaC 2010 — Algebra (Division A) — Problem 3

Let S be the sum of all real x such that 4 = x . Find the nearest integer to S . √ √ √ √

解析

Find the nearest integer to the sum of all x where 4 = x . Solution: 5. We immediately see two solutions, 2 and 4, and that there can be no more positive x 4 roots. There must be a negative root, however: let f ( x ) = 4 and g ( x ) = x ; then g (0) = 0 and f (0) = 1, but g goes off to infinity as x → −∞ and f goes to 0 as x → ∞ . Plugging in x = − 1, we have f ( − 1) = 1 / 4 and g ( x ) = 1; plugging in x = − 1 / 2 we have f ( − 1 / 2) = 1 / 2 and g ( x ) = 1 / 16. Therefore the root is between − 1 / 2 and − 1, and the nearest integer to the sum of the roots must be 5. √ √ √ √