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2015 SDMO (High School)
5
5
Part of
2015 SDMO (High School)
Problems
(1)
Always a 0.125-good point
Source: 2015 SDMO High School Problem 5
8/24/2016
Let
A
A
A
be a finite set of points in the coordinate plane. Suppose that
A
A
A
has
n
≥
3
n\geq3
n
≥
3
points. Given any
a
a
a
in
A
A
A
, the horizontal and vertical lines through
a
a
a
define four closed quadrants centered at
a
a
a
. For any real number
α
\alpha
α
, call a point
a
a
a
in
A
A
A
α
\alpha
α
-good if there are two diagonally opposite closed quadrants centered at
a
a
a
that each contain at least
α
n
\alpha n
α
n
points from
A
A
A
. Show that there is some
a
a
a
in
A
A
A
that is
1
8
\frac{1}{8}
8
1
-good.