HMMT 二月 2016 · 几何 · 第 7 题

HMMT February 2016 — Geometry — Problem 7

Let S = { ( x, y ) | x, y ∈ Z , 0 ≤ x, y, ≤ 2016 } . Given points A = ( x , y ) , B = ( x , y ) in S , define 1 1 2 2 2 2 d ( A, B ) = ( x − x ) + ( y − y ) (mod 2017) . 2017 1 2 1 2 The points A = (5 , 5) , B = (2 , 6) , C = (7 , 11) all lie in S . There is also a point O ∈ S that satisfies d ( O, A ) = d ( O, B ) = d ( O, C ) . 2017 2017 2017 Find d ( O, A ) . 2017 π

解析

Let S = { ( x, y ) | x, y ∈ Z , 0 ≤ x, y, ≤ 2016 } . Given points A = ( x , y ) , B = ( x , y ) in S , define 1 1 2 2 2 2 d ( A, B ) = ( x − x ) + ( y − y ) (mod 2017) . 2017 1 2 1 2 The points A = (5 , 5) , B = (2 , 6) , C = (7 , 11) all lie in S . There is also a point O ∈ S that satisfies d ( O, A ) = d ( O, B ) = d ( O, C ) . 2017 2017 2017 Find d ( O, A ) . 2017 Proposed by: Yang Liu Answer: 1021 2 2 (7 − 2) +(11 − 6) 25 2 Note that the triangle is a right triangle with right angle at A . Therefore, R = = = 4 2 − 1 (25)(2 ) ≡ 1021 (mod 2017). (An equivalent approach works for general triangles; the fact that the triangle is right simply makes the circumradius slightly easier to compute.) π