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Contests
National and Regional Contests
Poland Contests
Polish MO Finals
1962 Polish MO Finals
1962 Polish MO Finals
Part of
Polish MO Finals
Subcontests
(6)
6
1
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parallelepiped construction
Given three lines
a
a
a
,
b
b
b
,
c
c
c
pairwise skew. Is it possible to construct a parallelepiped whose edges lie on the lines
a
a
a
,
b
b
b
,
c
c
c
?
5
1
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\sqrt[n + 1]{n+1} < \sqrt[n]{n}.
Prove that if
n
n
n
is a natural number greater than
2
2
2
, then
n
+
1
n
+
1
<
n
n
.
\sqrt[n + 1]{n+1} < \sqrt[n]{n}.
n
+
1
n
+
1
<
n
n
.
4
1
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ways to partition a set of n items be into 2 sets
How many ways can a set of
n
n
n
items be partitioned into two sets?
3
1
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A-bisector , B-median and C- altitude concur
What condition should the angles of triangle
A
B
C
ABC
A
BC
satisfy so that the bisector of angle
A
A
A
, the median drawn from vertex
B
B
B
and the altitude drawn from vertex
C
C
C
intersect at one point?
2
1
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point construction, area of convex quad in 4 equal parts
Inside a given convex quadrilateral, find a point such that the segments connecting this point with the midpoints of the quadrilateral's sides divide the quadrilateral into four parts with equal areas.
1
1
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sum 1/a_k a_{k+1}=(n-1)/a_1a_n
Prove that if the numbers
a
1
,
a
2
,
…
,
a
n
a_1, a_2,\ldots, a_n
a
1
,
a
2
,
…
,
a
n
(
n
n
n
- natural number
≥
2
\geq 2
≥
2
) form an arithmetic progression, and none of them is zero, then
1
a
1
a
2
+
1
a
2
a
3
+
…
+
1
a
n
−
1
a
n
=
n
−
1
a
1
a
n
.
\frac{1}{a_1a_2} + \frac{1}{a_2a_3} + \ldots + \frac{1}{a_{n-1}a_n} = \frac{n-1}{a_1a_n}.
a
1
a
2
1
+
a
2
a
3
1
+
…
+
a
n
−
1
a
n
1
=
a
1
a
n
n
−
1
.