HMMT 二月 2011 · 冲刺赛 · 第 15 题

英文解析

15. [ 10 ] Find all irrational numbers x such that x − 17 x and x + 4 x are both rational numbers.
    √
    Answer: − 2 ± 5
    Guts Round
    √
    2 2 2
    From x + 4 x ∈ Q , we deduce that ( x + 2) = x + 4 x + 4 is also rational, and hence x = − 2 ± y ,
    √
    √
    where y is rational. Then x − 17 x = (26 − 6 y ) ± ( y − 5) y , which forces y to be 5. Hence x = − 2 ± 5.3
    It is easy to check that both values satisfy the problem conditions.