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HMMT 二月 2013 · 冲刺赛 · 第 1 题

HMMT February 2013 — Guts Round — Problem 1

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 4 ] Arpon chooses a positive real number k . For each positive integer n , he places a marker at the point
    ( n, nk ) in the ( x, y ) plane. Suppose that two markers whose x coordinates differ by 4 have distance
解析

英文解析

  1. Let B have coordinate b and C have coordinate c . We obtain easily that B is b , C is c , and D
    2 2
    is bc . Therefore, H is 1 + b + c and H is b + c + bc (we have used the fact that for triangles on the
    1 2
    1 2 2
    unit circle, their orthocenter is the sum of the vertices). Finally, we have that M is (1 + b + c ), so
    1 2 2 1 22
    the reflection of O about the midpoint of M H is (1 + b + c + 2 b + 2 c + 2 bc ) = ( b + c + 1) , so we
    2 22
    2 ° °1
    just seek | b + c + 1 | . But we know that b = cis 135 and c = cis 195 , so we obtain that this value is
    √ √2
    (8 − 6 − 3 2).1 4