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Problems
Contests
National and Regional Contests
Poland Contests
Poland - Second Round
1956 Poland - Second Round
1956 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(6)
6
1
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tetrahedron with AB x CD = AC x BD = AD x BC
Prove that if in a tetrahedron
A
B
C
D
ABCD
A
BC
D
the segments connecting the vertices of the tetrahedron with the centers of circles inscribed in opposite faces intersect at one point, then
A
B
⋅
C
D
=
A
C
⋅
B
D
=
A
D
⋅
B
C
AB \cdot CD = AC \cdot BD = AD \cdot BC
A
B
⋅
C
D
=
A
C
⋅
B
D
=
A
D
⋅
BC
and that the converse also holds.
5
1
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\sqrt{A} + \sqrt{B} + \sqrt{C} < 4 \sqrt{3}
Prove that the numbers
A
A
A
,
B
B
B
,
C
C
C
defined by the formulas
A
=
t
g
β
t
g
γ
+
5
,
B
=
t
g
γ
t
g
α
+
5
,
C
=
t
g
α
t
g
β
+
5
,
A = tg \beta tg \gamma + 5,\\ B = tg \gamma tg \alpha + 5,\\ C = tg \alpha tg \beta + 5,
A
=
t
g
βt
g
γ
+
5
,
B
=
t
g
γ
t
gα
+
5
,
C
=
t
gα
t
g
β
+
5
,
where
α
>
0
\alpha>0
α
>
0
,
β
>
0
\beta > 0
β
>
0
,
γ
>
0
\gamma > 0
γ
>
0
and
α
+
β
+
γ
=
9
0
∘
\alpha + \beta + \gamma = 90^\circ
α
+
β
+
γ
=
9
0
∘
, satisfy the inequality
A
+
B
+
C
<
4
3
.
\sqrt{A} + \sqrt{B} + \sqrt{C} < 4 \sqrt{3}.
A
+
B
+
C
<
4
3
.
4
1
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2x^2 - 215y^2 = 1 diophantine
Prove that the equation
2
x
2
−
215
y
2
=
1
2x^2 - 215y^2 = 1
2
x
2
−
215
y
2
=
1
has no integer solutions.
3
1
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uniform horizontal circular plate of weight
A uniform horizontal circular plate of weight
Q
Q
Q
kG is supported at points
A
A
A
,
B
B
B
,
C
C
C
lying on the circumference of the plate, with
A
C
=
B
C
AC = BC
A
C
=
BC
and
A
C
B
=
2
α
ACB = 2\alpha
A
CB
=
2
α
. What weight
x
x
x
kG must be placed on the plate at the other end
D
D
D
of the diameter drawn from point
C
C
C
so that the pressure of the plate on the support at
C
C
C
G is equal to zero?
2
1
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4 equal circumcircles, orthocenter
Prove that if
H
H
H
is the point of intersection of the altitudes of a non-right triangle
A
B
C
ABC
A
BC
, then the circumcircles of the triangles
A
H
B
AHB
A
H
B
,
B
H
C
BHC
B
H
C
,
C
H
A
CHA
C
H
A
and
A
B
C
ABC
A
BC
are equal.
1
1
Hide problems
x^3 + y^3 + z^3 + mxyz divisible by x + y + z
For what value of
m
m
m
is the polynomial
x
3
+
y
3
+
z
3
+
m
x
y
z
x^3 + y^3 + z^3 + mxyz
x
3
+
y
3
+
z
3
+
m
x
yz
divisible by
x
+
y
+
z
x + y + z
x
+
y
+
z
?