Library / /980 of 1394

, 2016

Geometry Difficulty 5.5 AIME, harder Prove it United States

Problem:

Let $S=\{(x, y) \mid x, y \in \mathbb{Z}, 0 \leq x, y \leq 2016\}$ . Given points $A=(x_1, y_1)$ , $B=(x_2, y_2)$ in $S$ , define
$d_{2017}(A, B) = (x_1 - x_2)^2 + (y_1 - y_2)^2 \pmod{2017}$

The points $A=(5,5)$ , $B=(2,6)$ , $C=(7,11)$ all lie in $S$ . There is also a point $O \in S$ that satisfies
$d_{2017}(O, A) = d_{2017}(O, B) = d_{2017}(O, C)$

Find $d_{2017}(O, A)$ .

Solution

Solution:

Note that the triangle is a right triangle with right angle at $A$ . Therefore,
$R^2 = \frac{(7-2)^2 + (11-6)^2}{4} = \frac{25}{2} = (25)\left(2^{-1}\right) \equiv 1021 \pmod{2017}.$
(An equivalent approach works for general triangles; the fact that the triangle is right simply makes the circumradius slightly easier to compute.)

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.