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National and Regional Contests
Poland Contests
Poland - Second Round
1955 Poland - Second Round
1955 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(6)
6
1
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1/ OM + 1/ ON + 1/OP is fixed, trihedral angle related
Inside the trihedral angle
O
A
B
C
OABC
O
A
BC
, whose plane angles
A
O
B
AOB
A
OB
,
B
O
C
BOC
BOC
,
C
O
A
COA
CO
A
are equal, a point
S
S
S
is chosen equidistant from the faces of this angle. Through point
S
S
S
a plane is drawn that intersects the edges
O
A
OA
O
A
,
O
B
OB
OB
,
O
C
OC
OC
at points
M
M
M
,
N
N
N
,
P
P
P
, respectively. Prove that the sum
1
O
M
+
1
O
N
+
1
O
P
\frac{1}{OM} + \frac{1}{ON} + \frac{1}{OP}
OM
1
+
ON
1
+
OP
1
has a constant value, i.e. independent of the position of the plane
M
N
P
MNP
MNP
.
5
1
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rectangle of smallest area containing the triangle.
Given a triangle
A
B
C
ABC
A
BC
. Find the rectangle of smallest area containing the triangle.
4
1
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broken line APQ bisects area of triangle, fixed point P inside ABC
Inside the triangle
A
B
C
ABC
A
BC
a point
P
P
P
is given; find a point
Q
Q
Q
on the perimeter of this triangle such that the broken line
A
P
Q
APQ
A
PQ
divides the triangle into two parts with equal areas.
3
1
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triangle construction
What should the angle at the vertex of an isosceles triangle be so that it is possible to construct a triangle with sides equal to the height, base, and one of the other sides of the isosceles triangle?
2
1
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1 + 2 + 3 + .. + n = three-digit number with identical digits.
Find the natural number
n
n
n
knowing that the sum
1
+
2
+
3
+
…
+
n
1 + 2 + 3 + \ldots + n
1
+
2
+
3
+
…
+
n
is a three-digit number with identical digits.
1
1
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x^4 + y^4 + z^4=? if x + y + z = 0 and x^2 + y^2 + z^2 = a
Calculate the sum
x
4
+
y
4
+
z
4
x^4 + y^4 + z^4
x
4
+
y
4
+
z
4
knowing that
x
+
y
+
z
=
0
x + y + z = 0
x
+
y
+
z
=
0
and
x
2
+
y
2
+
z
2
=
a
x^2 + y^2 + z^2 = a
x
2
+
y
2
+
z
2
=
a
, where
a
a
a
is a given positive number.