MathDB
Problems
Contests
Undergraduate contests
Miklós Schweitzer
1993 Miklós Schweitzer
1
1
Part of
1993 Miklós Schweitzer
Problems
(1)
geometry
Source: miklos schweitzer 1993 q1
10/22/2021
There are n points in the plane with the property that the distance between any two points is at least 1. Prove that for sufficiently large n , the number of pairs of points whose distance is in
[
t
1
,
t
1
+
1
]
∪
[
t
2
,
t
2
+
1
]
[ t_1 , t_1 + 1] \cup [ t_2 , t_2 + 1]
[
t
1
,
t
1
+
1
]
∪
[
t
2
,
t
2
+
1
]
for some
t
1
,
t
2
t_1, t_2
t
1
,
t
2
, is at most
[
n
2
3
]
[\frac{n^2}{3}]
[
3
n
2
]
, and the bound is sharp.
geometry
combinatorics