HMMT 二月 2013 · 冲刺赛 · 第 1 题

HMMT February 2013 — Guts Round — Problem 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

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 2 1 2 2 1 2 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 2 2 1 2 ◦ ◦ just seek | b + c + 1 | . But we know that b = cis135 and c = cis195 , so we obtain that this value is 2 √ √ 1 (8 − 6 − 3 2). 4