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Problems
Contests
National and Regional Contests
Poland Contests
Poland - Second Round
1960 Poland - Second Round
1960 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(6)
6
1
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volume of the tetrahedron
Calculate the volume of the tetrahedron
A
B
C
D
ABCD
A
BC
D
given the edges
A
B
=
b
AB = b
A
B
=
b
,
A
C
=
c
AC = c
A
C
=
c
,
A
D
=
d
AD = d
A
D
=
d
and the angles
∡
C
A
D
=
β
\measuredangle CAD = \beta
∡
C
A
D
=
β
,
∡
D
A
B
=
γ
\measuredangle DAB = \gamma
∡
D
A
B
=
γ
and
∡
B
A
C
=
δ
\measuredangle BAC = \delta
∡
B
A
C
=
δ
.
5
1
Hide problems
concyclic
There are three different points on the line
A
A
A
,
B
B
B
,
C
C
C
and a point
S
S
S
outside this line; perpendicularly drawn at points
A
A
A
,
B
B
B
,
C
C
C
to the lines
S
A
SA
S
A
,
S
B
SB
SB
,
S
C
SC
SC
intersect at points
M
M
M
,
N
N
N
,
P
P
P
. Prove that the points
M
M
M
,
N
N
N
,
P
P
P
,
S
S
S
lie on the circle
4
1
Hide problems
7 | 2^{n+2} + 3^{2n+1}
Prove that if
n
n
n
is a non-negative integer, then number
2
n
+
2
+
3
2
n
+
1
2^{n+2} + 3^{2n+1}
2
n
+
2
+
3
2
n
+
1
is divisible by
7
7
7
.
3
1
Hide problems
collinear wanted, 2 concentric circles
There are two circles with a common center
O
O
O
and a point
A
A
A
. Construct a circle with center
A
A
A
intersecting the given circles at points
M
M
M
and
N
N
N
such that the line
M
N
MN
MN
passes through point
O
O
O
.
2
1
Hide problems
x^2 + p_1x + q_1 = 0, x^2 + p_2x + q_2 = 0, x^2 + p_3x + q_3 = 0,
The equations are given
x
2
+
p
1
x
+
q
1
=
0
x
2
+
p
2
x
+
q
2
=
0
x
2
+
p
3
x
+
q
3
=
0
\begin{array}{c} x^2 + p_1x + q_1 = 0\\ x^2 + p_2x + q_2 = 0\\ x^2 + p_3x + q_3 = 0 \end{array}
x
2
+
p
1
x
+
q
1
=
0
x
2
+
p
2
x
+
q
2
=
0
x
2
+
p
3
x
+
q
3
=
0
each two of which have a common root, but all three have no common root. Prove that:1)
2
(
p
1
p
2
+
p
2
p
3
+
p
3
p
1
)
−
(
p
1
2
+
p
2
2
+
p
3
2
)
=
4
(
q
1
+
q
2
+
q
3
)
2 (p_1p_2 + p_2p_3 + p_3p_1) - (p_1^2 + p_2^2 + p_3^2) = 4 (q_1 + q_2+ q_3)
2
(
p
1
p
2
+
p
2
p
3
+
p
3
p
1
)
−
(
p
1
2
+
p
2
2
+
p
3
2
)
=
4
(
q
1
+
q
2
+
q
3
)
2) he roots of these equations are rational when the numbers
p
1
p_1
p
1
,
p
2
p_2
p
2
and
p
3
p_3
p
3
are rational}.
1
1
Hide problems
a^{2n} + a^{2n-1}b + a^{2n-2} b^2 + \ldots + ab^{2n-1} + b^{2n} > 0
Prove that if the real numbers
a
a
a
and
b
b
b
are not both equal to zero, then for every natural
n
n
n
a
2
n
+
a
2
n
−
1
b
+
a
2
n
−
2
b
2
+
…
+
a
b
2
n
−
1
+
b
2
n
>
0.
a^{2n} + a^{2n-1}b + a^{2n-2} b^2 + \ldots + ab^{2n-1} + b^{2n} > 0.
a
2
n
+
a
2
n
−
1
b
+
a
2
n
−
2
b
2
+
…
+
a
b
2
n
−
1
+
b
2
n
>
0.