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Problems
Contests
National and Regional Contests
Mexico Contests
Mexican Girls' Contest
2022 Mexican Girls' Contest
8
8
Part of
2022 Mexican Girls' Contest
Problems
(1)
Marking vertices in splitted triangle
Source: Mexico
2/7/2022
Let
n
n
n
be a positive integer. Consider a figure of a equilateral triangle of side
n
n
n
and splitted in
n
2
n^2
n
2
small equilateral triangles of side
1
1
1
. One will mark some of the
1
+
2
+
⋯
+
(
n
+
1
)
1+2+\dots+(n+1)
1
+
2
+
⋯
+
(
n
+
1
)
vertices of the small triangles, such that for every integer
k
≥
1
k\geq 1
k
≥
1
, there is not any trapezoid(trapezium), whose the sides are
(
1
,
k
,
1
,
k
+
1
)
(1,k,1,k+1)
(
1
,
k
,
1
,
k
+
1
)
, with all the vertices marked. Furthermore, there are no small triangle(side
1
1
1
) have your three vertices marked. Determine the greatest quantity of marked vertices.
geometry
trapezoid