HMMT 二月 2008 · 冲刺赛 · 第 17 题
HMMT February 2008 — Guts Round — Problem 17
题目详情
英文原题
- [ 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 primefactorization.
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解析
英文解析
- [ 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 primefactorization.
Answer: Rewrite the equation to get 1
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 theoriginal equation.
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