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Problems
Contests
National and Regional Contests
Poland Contests
Polish MO Finals
1955 Polish MO Finals
1955 Polish MO Finals
Part of
Polish MO Finals
Subcontests
(6)
6
1
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computational geo with a sphere
Through points
A
A
A
and
B
B
B
two oblique lines
m
m
m
and
n
n
n
are drawn perpendicular to the line
A
B
AB
A
B
. On line
m
m
m
the point
C
C
C
(different from
A
A
A
) is taken, and on line
n
n
n
the point
D
D
D
(different from
B
B
B
) is taken. Given the lengths of segments
A
B
=
d
AB = d
A
B
=
d
and
C
D
=
l
CD = l
C
D
=
l
and the angle
φ
\varphi
φ
formed by the oblique lines
m
m
m
and
n
n
n
, calculate the radius of the surface of the sphere passing through the points
A
A
A
,
B
B
B
,
C
C
C
,
D
D
D
.
5
1
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max distance |MA-MB|
In the plane, a straight line
m
m
m
is given and points
A
A
A
and
B
B
B
lie on opposite sides of the straight line
m
m
m
. Find a point
M
M
M
on the line
m
m
m
such that the difference in distances of this point from points
A
A
A
and
B
B
B
is as large as possible.
4
1
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sin^2 a + sin^2 b >=sin a sin b+ sin a+ sinb -1
Prove that
sin
2
α
+
sin
2
β
≥
sin
α
sin
β
+
sin
α
+
sin
β
−
1.
\sin^2 \alpha + \sin^2 \beta \geq \sin \alpha \sin \beta + \sin \alpha + \sin \beta - 1.
sin
2
α
+
sin
2
β
≥
sin
α
sin
β
+
sin
α
+
sin
β
−
1.
3
1
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MA=MB+MC, equilateral ABC inscribed, scent of a classic
An equilateral triangle
A
B
C
ABC
A
BC
is inscribed in a circle; prove that if
M
M
M
is any point of the circle, then one of the distances
M
A
MA
M
A
,
M
B
MB
MB
,
M
C
MC
MC
is equal to the sum of the other two.
1
1
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x^3 + ax^2 + bx + c = 0
What conditions must the real numbers
a
a
a
,
b
b
b
, and
c
c
c
satisfy so that the equation
x
3
+
a
x
2
+
b
x
+
c
=
0
x^3 + ax^2 + bx + c = 0
x
3
+
a
x
2
+
b
x
+
c
=
0
has three distinct real roots forming a geometric progression?
2
1
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7 naturals an arithmetic progression with difference $ 30
Prove that among the seven natural numbers forming an arithmetic progression with difference
30
30
30
, one and only one is divisible by
7
7
7
.