MathDB

S

be the set of all

2017^2

lattice points

(x,y)

with

x,y\in \{0\}\cup\{2^{0},2^{1},\cdots,2^{2015}\}

. A subset

X\subseteq S

is called BQ if it has the following properties:
(a)

X

contains at least three points, no three of which are collinear. (b) One of the points in

X

(0,0)

. (c) For any three distinct points

A,B,C \in X

, the orthocenter of

\triangle ABC

is in

X

. (d) The convex hull of

X

contains at least one horizontal line segment.
Determine the number of BQ subsets of

S

.
Proposed by Vincent Huang

Problem Statement