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2004 District Olympiad
4
14 specific points on a ’chocolate’ tablet
14 specific points on a ’chocolate’ tablet
Source: Romanian District Olympiad, Grade IX, Problem 4
October 7, 2018
geometry
rectangle
vectorial geometry
analytic geometry
Problem Statement
Divide a
2
×
4
2\times 4
2
×
4
rectangle into
8
8
8
unit squares to obtain a set of
15
15
15
vertices denoted by
M
.
\mathcal{M} .
M
.
Find the points
A
∈
M
A\in\mathcal{M}
A
∈
M
that have the property that the set
M
∖
{
A
}
\mathcal{M}\setminus \{ A\}
M
∖
{
A
}
can form
7
7
7
pairs
(
A
1
,
B
1
)
,
(
A
2
,
B
2
)
,
…
,
(
A
7
,
B
7
)
∈
M
×
M
\left( A_1,B_1\right) ,\left( A_2,B_2\right) ,\ldots ,\left( A_7,B_7\right)\in\mathcal{M}\times\mathcal{M}
(
A
1
,
B
1
)
,
(
A
2
,
B
2
)
,
…
,
(
A
7
,
B
7
)
∈
M
×
M
such that
A
1
B
1
→
+
A
2
B
2
→
+
⋯
+
A
7
B
7
→
=
O
→
.
\overrightarrow{A_1B_1} +\overrightarrow{A_2B_2} +\cdots +\overrightarrow{A_7B_7} =\overrightarrow{O} .
A
1
B
1
+
A
2
B
2
+
⋯
+
A
7
B
7
=
O
.
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