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Romanian Masters of Mathematics Collection
2023 Romanian Master of Mathematics
2
2
Part of
2023 Romanian Master of Mathematics
Problems
(1)
Beautiful Triangles
Source: 2023 RMM, Problem 2
3/1/2023
Fix an integer
n
≥
3
n \geq 3
n
≥
3
. Let
S
\mathcal{S}
S
be a set of
n
n
n
points in the plane, no three of which are collinear. Given different points
A
,
B
,
C
A,B,C
A
,
B
,
C
in
S
\mathcal{S}
S
, the triangle
A
B
C
ABC
A
BC
is nice for
A
B
AB
A
B
if
[
A
B
C
]
≤
[
A
B
X
]
[ABC] \leq [ABX]
[
A
BC
]
≤
[
A
BX
]
for all
X
X
X
in
S
\mathcal{S}
S
different from
A
A
A
and
B
B
B
. (Note that for a segment
A
B
AB
A
B
there could be several nice triangles). A triangle is beautiful if its vertices are all in
S
\mathcal{S}
S
and is nice for at least two of its sides.Prove that there are at least
1
2
(
n
−
1
)
\frac{1}{2}(n-1)
2
1
(
n
−
1
)
beautiful triangles.
combinatorics
RMM 2023