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Contests
National and Regional Contests
Austria Contests
Austrian MO Regional Competition
1970 Regional Competition For Advanced Students
1970 Regional Competition For Advanced Students
Part of
Austrian MO Regional Competition
Subcontests
(4)
4
1
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Find real solutions to set of equations
Find all real solutions of the following set of equations:
72
x
3
+
4
x
y
2
=
11
y
3
72x^3+4xy^2=11y^3
72
x
3
+
4
x
y
2
=
11
y
3
27
x
5
−
45
x
4
y
−
10
x
2
y
3
=
−
143
32
y
5
27x^5-45x^4y-10x^2y^3=\frac{-143}{32}y^5
27
x
5
−
45
x
4
y
−
10
x
2
y
3
=
32
−
143
y
5
3
1
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Distance between points is p, intersection between E and P
E
1
E_1
E
1
and
E
2
E_2
E
2
are parallel planes and their distance is
p
p
p
. (a) How long is the seitenkante of the regular octahedron such that a side lies in
E
1
E_1
E
1
and another in
E
2
E_2
E
2
? (b)
E
E
E
is a plane between
E
1
E_1
E
1
and
E
2
E_2
E
2
, parallel to
E
1
E_1
E
1
and
E
2
E_2
E
2
, so that its distances from
E
1
E_1
E
1
and
E
2
E_2
E
2
are in ratio
1
:
2
1:2
1
:
2
Draw the intersection figure of
E
E
E
and the octahedron for
P
=
4
3
2
P=4\sqrt{\frac32}
P
=
4
2
3
cm and justifies, why the that figure must look in such a way
2
1
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Maximum number of intersections of all perpendiculars to 286
In the plane seven different points
P
1
,
P
2
,
P
3
,
P
4
,
Q
1
,
Q
2
,
Q
3
P_1, P_2, P_3, P_4, Q_1, Q_2, Q_3
P
1
,
P
2
,
P
3
,
P
4
,
Q
1
,
Q
2
,
Q
3
are given. The points
P
1
,
P
2
,
P
3
,
P
4
P_1, P_2, P_3, P_4
P
1
,
P
2
,
P
3
,
P
4
are on the straight line
p
p
p
, the points
Q
1
,
Q
2
,
Q
3
Q_1, Q_2, Q_3
Q
1
,
Q
2
,
Q
3
are not on
p
p
p
. By each of the three points
Q
1
,
Q
2
,
Q
3
Q_1, Q_2, Q_3
Q
1
,
Q
2
,
Q
3
the perpendiculars are drawn on the straight lines connecting points different of them. Prove that the maximum's number of the possibles intersections of all perpendiculars is to 286, if the points
Q
1
,
Q
2
,
Q
3
Q_1, Q_2, Q_3
Q
1
,
Q
2
,
Q
3
are taken in account as intersections.
1
1
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1+1/x+1+1/y+1+1/z\ge 64 when x+y+z=1
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be positive real numbers such that
x
+
y
+
z
=
1
x+y+z=1
x
+
y
+
z
=
1
Prove that always
(
1
+
1
x
)
×
(
1
+
1
y
)
×
(
1
+
1
z
)
≥
64
\left( 1+\frac1x\right)\times\left(1+\frac1y\right)\times\left(1 +\frac1z\right)\ge 64
(
1
+
x
1
)
×
(
1
+
y
1
)
×
(
1
+
z
1
)
≥
64
When does equality hold?