MathDB
Problems
Contests
National and Regional Contests
Romania Contests
Romania Team Selection Test
1994 BMO TST – Romania
1994 BMO TST – Romania
Part of
Romania Team Selection Test
Subcontests
(4)
3:
1
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intressting comb problem
Let
M
1
,
M
2
,
.
.
.
,
M
11
M_1, M_2, . . ., M_{11}
M
1
,
M
2
,
...
,
M
11
be
5
−
5-
5
−
element sets such that M_i \cap M_j \neq {\O} for all
i
,
j
∈
{
1
,
.
.
.
,
11
}
i, j \in \{1, . . ., 11\}
i
,
j
∈
{
1
,
...
,
11
}
. Determine the minimum possible value of the greatest number of the given sets that have nonempty intersection.
4:
1
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combinatorics,problem
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
)
2:
1
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combinatorics problem
Let
n
≥
4
n\geq 4
n
≥
4
be an integer. Find the maximum possible area of an
n
−
g
o
n
n-gon
n
−
g
o
n
inscribed in a unit cicle and having two perpendicular diagonals.
1:
1
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problem 1:bmo TST
Prove that if
n
n
n
is a square-free positive integer, there are no coprime positive integers
x
x
x
and
y
y
y
such that
(
x
+
y
)
3
(x + y)^3
(
x
+
y
)
3
divides
x
n
+
y
n
x^n+y^n
x
n
+
y
n