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Poland - First Round
1992 Poland - First Round
7
Polyhedra and an overrightarrow equation.
Polyhedra and an overrightarrow equation.
Source: Poland Math Olympiad 1992 First Round #7
June 2, 2023
LaTeX
Problem Statement
Given are the points
A
0
=
(
0
,
0
,
0
)
,
A
1
=
(
1
,
0
,
0
)
,
A
2
=
(
0
,
1
,
0
)
,
A
3
=
(
0
,
0
,
1
)
A_0 = (0,0,0), A_1 = (1,0,0), A_2 = (0,1,0), A_3 = (0,0,1)
A
0
=
(
0
,
0
,
0
)
,
A
1
=
(
1
,
0
,
0
)
,
A
2
=
(
0
,
1
,
0
)
,
A
3
=
(
0
,
0
,
1
)
in the space. Let
P
i
j
(
i
,
j
∈
0
,
1
,
2
,
3
)
P_{ij} (i,j \in 0,1,2,3)
P
ij
(
i
,
j
∈
0
,
1
,
2
,
3
)
be the point determined by the equality:
A
0
P
i
j
→
=
A
i
A
j
→
\overrightarrow{A_0P_{ij}} = \overrightarrow{A_iA_j}
A
0
P
ij
=
A
i
A
j
. Find the volume of the smallest convex polyhedron which contains all the points
P
i
j
P_{ij}
P
ij
.
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