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Problems
Contests
National and Regional Contests
Poland Contests
Poland - Second Round
1974 Poland - Second Round
1974 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(6)
6
1
Hide problems
all terms of the sequence are equal.
There is a sequence of integers
a
1
,
a
2
,
…
,
a
2
n
+
1
a_1, a_2, \ldots, a_{2n+1}
a
1
,
a
2
,
…
,
a
2
n
+
1
with the following property: after eliminating any term, the remaining ones can be divided into two groups of
n
n
n
terms such that the sum of the terms in the first group is equal to the sum words in the second. Prove that all terms of the sequence are equal.
5
1
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t = lim sum_{j=1}^N q^{n_j}.
The given numbers are real numbers
q
,
t
∈
⟨
1
2
;
1
)
q,t \in \langle \frac{1}{2}; 1)
q
,
t
∈
⟨
2
1
;
1
)
,
t
∈
(
0
;
1
⟩
t \in (0; 1 \rangle
t
∈
(
0
;
1
⟩
. Prove that there is an increasing sequence of natural numbers
n
k
{n_k}
n
k
(
k
=
1
,
2
,
…
k = 1,2, \ldots
k
=
1
,
2
,
…
) such that
t
=
lim
N
→
∞
∑
j
=
1
N
q
n
j
.
t = \lim_{N\to \infty} \sum_{j=1}^N q^{n_j}.
t
=
N
→
∞
lim
j
=
1
∑
N
q
n
j
.
4
1
Hide problems
sum of areas, 9 parts in a quad
In a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
with area
S
S
S
, each side was divided into 3 equal parts and segments were drawn connecting the appropriate points of division of the opposite sides in such a way that the quadrilateral was divided into 9 quadrilaterals. Prove that the sum of the areas of the following three quadrilaterals resulting from the division: the one containing the vertex
A
A
A
, the middle one and the one containing the vertex
C
C
C
is equal to
S
3
\frac{S}{3}
3
S
.
3
1
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coplanar projections of a vertex of tetrahedron on angle bisectors
Prove that the orthogonal projections of the vertex
D
D
D
of the tetrahedron
A
B
C
D
ABCD
A
BC
D
onto the bisectors of the internal and external dihedral angles at the edges
A
B
‾
\overline{AB}
A
B
,
B
C
‾
\overline{BC}
BC
and
C
A
‾
\overline{CA}
C
A
belong to one plane .
2
1
Hide problems
double sum t_it_j |s_i-s_j| < = 0
Prove that for every
n
=
2
,
3
,
…
n = 2, 3, \ldots
n
=
2
,
3
,
…
and any real numbers
t
1
,
t
2
,
…
,
t
n
t_1, t_2, \ldots, t_n
t
1
,
t
2
,
…
,
t
n
,
s
1
,
s
2
,
…
,
s
n
s_1, s_2, \ldots, s_n
s
1
,
s
2
,
…
,
s
n
, if
∑
i
=
1
n
t
i
=
0
,
to
∑
i
=
1
n
∑
j
=
1
n
t
i
t
j
∣
s
i
−
s
j
∣
≤
0.
\sum_{i=1}^n t_i = 0, \text{ to } \sum_{i=1}^n\sum_{j=1}^n t_it_j |s_i-s_j| \leq 0.
i
=
1
∑
n
t
i
=
0
,
to
i
=
1
∑
n
j
=
1
∑
n
t
i
t
j
∣
s
i
−
s
j
∣
≤
0.
1
1
Hide problems
pairs in a set
Let
Z
Z
Z
be a set of
n
n
n
elements. Find the number of such pairs of sets
(
A
,
B
)
(A, B)
(
A
,
B
)
such that
A
A
A
is contained in
B
B
B
and
B
B
B
is contained in
Z
Z
Z
. We assume that every set also contains itself and the empty set.