MathDB
Problems
Contests
National and Regional Contests
Poland Contests
Poland - Second Round
1971 Poland - Second Round
1971 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(6)
2
1
Hide problems
1 < cos A + cos B + cos C <= 3/2
Prove that if
A
,
B
,
C
A, B, C
A
,
B
,
C
are angles of a triangle, then
1
<
cos
A
+
cos
B
+
cos
C
≤
3
2
.
1 < \cos A + \cos B + \cos C \leq \frac{3}{2}.
1
<
cos
A
+
cos
B
+
cos
C
≤
2
3
.
6
1
Hide problems
(a_1+a_3+\ldots a_{2n+1})/(n+1) >= (a_2+a_4+\ldots a_{2n})/ n
Given an infinite sequence
{
a
n
}
\{a_n\}
{
a
n
}
. Prove that if
a
n
+
a
n
+
2
>
2
a
n
+
1
f
o
r
n
=
1
,
2...
a_n + a_{n+2} > 2a_{n+1} \ \ for \ \ n = 1, 2 ...
a
n
+
a
n
+
2
>
2
a
n
+
1
f
or
n
=
1
,
2...
then
a
1
+
a
3
+
…
a
2
n
+
1
n
+
1
≥
a
2
+
a
4
+
…
a
2
n
n
\frac{a_1+a_3+\ldots a_{2n+1}}{n+1} \geq \frac{a_2+a_4+\ldots a_{2n}}{n}
n
+
1
a
1
+
a
3
+
…
a
2
n
+
1
≥
n
a
2
+
a
4
+
…
a
2
n
for
n
=
1
,
2
,
…
n = 1, 2, \ldots
n
=
1
,
2
,
…
.
5
1
Hide problems
max sum prod a_{i,j}
Given the set of numbers
{
1
,
2
,
3
,
…
,
100
}
\{1, 2, 3, \ldots, 100\}
{
1
,
2
,
3
,
…
,
100
}
. From this set, create 10 pairwise disjoint subsets
N
i
=
{
a
i
,
1
,
a
i
,
2
,
.
.
.
a
i
,
10
N_i = \{a_{i,1}, a_{i,2}, ... a_{i,10}
N
i
=
{
a
i
,
1
,
a
i
,
2
,
...
a
i
,
10
(
i
=
1
,
2
,
…
,
10
i = 1, 2, \ldots, 10
i
=
1
,
2
,
…
,
10
) so that the sum of the products
∑
i
=
10
10
∏
j
=
1
10
a
i
,
j
\sum_{i=10}^{10}\prod_{j=1}^{10} a_{i,j}
i
=
10
∑
10
j
=
1
∏
10
a
i
,
j
was the biggest.
4
1
Hide problems
finite set of points, different distances
On the plane there is a finite set of points
Z
Z
Z
with the property that no two distances of the points of the set
Z
Z
Z
are equal. We connect the points
A
,
B
A, B
A
,
B
belonging to
Z
Z
Z
if and only if
A
A
A
is the point closest to
B
B
B
or
B
B
B
is the point closest to
A
A
A
. Prove that no point in the set
Z
Z
Z
will be connected to more than five others.
3
1
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among 6 lines in space here are 3 mutually oblique lines
There are 6 lines in space, of which no 3 are parallel, no 3 pass through the same point, and no 3 are contained in the same plane. Prove that among these 6 lines there are 3 mutually oblique lines.
1
1
Hide problems
k squares in a n x n chessboard
In how many ways can you choose
k
k
k
squares on a chessboard
n
×
n
n \times n
n
×
n
(
k
≤
n
k \leq n
k
≤
n
) so that no two of the chosen squares lie in the same row or column?