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National and Regional Contests
Poland Contests
Polish MO Finals
1953 Polish MO Finals
1953 Polish MO Finals
Part of
Polish MO Finals
Subcontests
(6)
6
1
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tan A+ tan B + tan C= tan A tan B tan C
What algebraic relationship holds between
α
\alpha
α
,
β
\beta
β
and
γ
\gamma
γ
when the equality is satisfied
tan
α
+
tan
β
+
tan
γ
=
tan
α
tan
β
tan
γ
?
\tan \alpha + \tan \beta + \tan \gamma = \tan \alpha \tan \beta \tan \gamma?
tan
α
+
tan
β
+
tan
γ
=
tan
α
tan
β
tan
γ
?
5
1
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minimum speed for cyclist
From point
O
O
O
a car starts on a straight road and travels with constant speed
v
v
v
. A cyclist who is located at a distance
a
a
a
from point
O
O
O
and at a distance
b
b
b
from the road wants to deliver a letter to this car. What is the minimum speed a cyclist should ride to reach his goal?
4
1
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(\sqrt{2}- 1)^n = \sqrt{m} - \sqrt{m-1}
Prove that if
n
n
n
is a natural number, then equality holds
(
2
−
1
)
n
=
m
−
m
−
1
(\sqrt{2}- 1)^n = \sqrt{m} - \sqrt{m-1}
(
2
−
1
)
n
=
m
−
m
−
1
where
m
m
m
is a natural number.
3
1
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volume of the tetrahedron formed by planes
Through each vertex of a tetrahedron with a given volume
V
V
V
, a plane is drawn parallel to the opposite face of the tetrahedron. Calculate the volume of the tetrahedron formed by these planes.
2
1
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locus of centers of rectangles inscribed in triangle
Find the geometric locus of the center of a rectangle whose vertices lie on the perimeter of a given triangle.
1
1
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1/(x - a) +1/(x - b)+ 1/(x - c)= 0
Test whether equation
1
x
−
a
+
1
x
−
b
+
1
x
−
c
=
0
,
\frac{1}{x - a} + \frac{1}{x - b} + \frac{1}{x - c} = 0,
x
−
a
1
+
x
−
b
1
+
x
−
c
1
=
0
,
where
a
a
a
,
b
b
b
,
c
c
c
denote the given real numbers, has real roots.