HMMT 二月 2016 · 几何 · 第 7 题
HMMT February 2016 — Geometry — Problem 7
题目详情
英文原题
- 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 satisfiesd ( O, A ) = d ( O, B ) = d ( O, C ) .
2017 2017 2017
Find d ( O, A ) .
2017 π
解析
英文解析
- 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 satisfiesd ( 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)
Note that the triangle is a right triangle with right angle at A . Therefore, R = = =225
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