HMMT 二月 2008 · 冲刺赛 · 第 17 题

HMMT February 2008 — Guts Round — Problem 17

[ 9 ] Solve the equation √ √ √ √ √ √ √ √ √ 2008 x + 4 x + 16 x + . . . + 4 x + 3 − x = 1 . Express your answer as a reduced fraction with the numerator and denominator written in their prime factorization. ◦

解析

[ 9 ] Solve the equation √ √ √ √ √ √ √ √ √ 2008 x + 4 x + 16 x + . . . + 4 x + 3 − x = 1 . Express your answer as a reduced fraction with the numerator and denominator written in their prime factorization. 1 Answer: Rewrite the equation to get 4016 2 √ √ √ √ √ √ √ √ √ 2008 x + 4 x + 16 x + . . . + 4 x + 3 = x + 1 . Squaring both sides yields √ √ √ √ 2008 4 x + . . . + 4 x + 3 = 2 x + 1 . Squaring again yields √ √ √ √ 2008 16 x + . . . + 4 x + 3 = 4 x + 1 . One can see that by continuing this process one gets √ √ 2008 2008 4 x + 3 = 2 x + 1 , √ 2008 − 2008 so that 2 · 2 x = 2. Hence x = 4 . It is also easy to check that this is indeed a solution to the original equation. ◦