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Contests
National and Regional Contests
Poland Contests
Poland - Second Round
1978 Poland - Second Round
1978 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(3)
5
1
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congruent tetrahedra
Prove that there is no inclined plane such that any tetrahedron placed arbitrarily with a certain face on the plane will not fall over. It means the following:Given a plane
π
\pi
π
and a line
l
l
l
not perpendicular to it. Prove that there is a tetrahedron
T
T
T
such that for each of its faces
S
S
S
there is in the plane
π
\pi
π
a triangle
A
B
C
ABC
A
BC
congruent to
S
S
S
and there is a point
D
D
D
such that the tetrahedron
A
B
C
D
ABCD
A
BC
D
congruent to
T
T
T
and the line parallel to
l
l
l
passing through the center of gravity of the tetrahedron
A
B
C
D
ABCD
A
BC
D
does not intersect the triangle
A
B
C
ABC
A
BC
.Note. The center of gravity of a tetrahedron is the intersection point of the segments connecting the centers of gravity of the faces of this tetrahedron with the opposite vertices (it is known that such a point always exists).
4
1
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limit of probability a triangle is acute, when vertices lie on regular 2n-gon
Three different points were randomly selected from the vertices of the regular
2
n
2n
2
n
-gon. Let
p
n
p_n
p
n
be the probability of the event that the triangle with vertices at the selected points is acute-angled. Calculate
lim
n
→
∞
p
n
\lim_{n\to \infty} p_n
lim
n
→
∞
p
n
.Attention. We assume that all choices of three different points are equally likely.
3
1
Hide problems
choose from sequence, a finite sequence with sum of terms = k
Given a sequence of natural numbers
(
a
i
)
(a_i)
(
a
i
)
, for each natural number
n
n
n
the sum of the terms of the sequence that are not greater than
n
n
n
is a number not less than
n
n
n
. Prove that for every natural number
k
k
k
it is possible to choose from the sequence
(
a
i
)
(a_i)
(
a
i
)
a finite sequence with the sum of terms equal to
k
k
k
.