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Romania Team Selection Test
1994 BMO TST – Romania
4:
4:
Part of
1994 BMO TST – Romania
Problems
(1)
combinatorics,problem
Source: Romania TST for BMO,P4
9/6/2017
Consider a tetrahedron
A
1
A
2
A
3
A
4
A_1A_2A_3A_4
A
1
A
2
A
3
A
4
. A point
N
N
N
is said to be a Servais point if its projections on the six edges of the tetrahedron lie in a plane
α
(
N
)
\alpha(N)
α
(
N
)
(called Servais plane). Prove that if all the six points
N
i
j
Nij
N
ij
symmetric to a point
M
M
M
with respect to the midpoints
B
i
j
Bij
B
ij
of the edges
A
i
A
j
A_iA_j
A
i
A
j
are Servais points, then
M
M
M
is contained in all Servais planes
α
(
N
i
j
)
\alpha(Nij )
α
(
N
ij
)
combinatorics