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Problems
Contests
National and Regional Contests
Poland Contests
Poland - Second Round
1969 Poland - Second Round
1969 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(6)
6
1
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every polyhedron has at least 2 faces with same number of sides.
Prove that every polyhedron has at least two faces with the same number of sides.
5
1
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parallel projection of one plane onto another plane,
Prove that if, in parallel projection of one plane onto another plane, the image of a certain square is a square, then the image of every figure is the figure congruent to it.
4
1
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1^m + 2^m + .. + n^m >= n\cdot ( \frac{n+1}{2})^m
Prove that for any natural numbers min the inequality holds
1
m
+
2
m
+
…
+
n
m
≥
n
⋅
(
n
+
1
2
)
m
1^m + 2^m + \ldots + n^m \geq n\cdot \left( \frac{n+1}{2}\right)^m
1
m
+
2
m
+
…
+
n
m
≥
n
⋅
(
2
n
+
1
)
m
3
1
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concyclic , given cyclic quad
Given a quadrilateral
A
B
C
D
ABCD
A
BC
D
inscribed in a circle. The images of the points
A
A
A
and
C
C
C
in symmetry with respect to the line
B
D
BD
B
D
are the points
A
′
A'
A
′
and
C
′
C'
C
′
, respectively, and the images of the points
B
B
B
and
D
D
D
in symmetry with respect to the line
A
C
AC
A
C
are the points
B
′
B'
B
′
and
D
′
D'
D
′
respectively. Prove that the points
A
′
A'
A
′
,
B
′
B'
B
′
,
C
′
C'
C
′
,
D
′
D'
D
′
lie on the circle.
2
1
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4-digit numbers wanted
Find all four-digit numbers in which the thousands digit is equal to the hundreds digit and the tens digit is equal to the units digit and which are squares of integers.
1
1
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a^2 + b^2 = 1, c^2 + d^2 = 1, ac + bd = - 1/2
Prove that if the real numbers
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
satisfy the equations
a
2
+
b
2
=
1
,
c
2
+
d
2
=
1
,
a
c
+
b
d
=
−
1
2
,
\; a^2 + b^2 = 1,\; c^2 + d^2 = 1, \; ac + bd = -\frac{1}{2},
a
2
+
b
2
=
1
,
c
2
+
d
2
=
1
,
a
c
+
b
d
=
−
2
1
,
then
a
2
+
a
c
+
c
2
=
b
2
+
b
d
+
d
2
.
a^2 + ac + c^2 = b^2 + bd + d^2.
a
2
+
a
c
+
c
2
=
b
2
+
b
d
+
d
2
.