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Problems
Contests
National and Regional Contests
Poland Contests
Polish MO Finals
1954 Polish MO Finals
1954 Polish MO Finals
Part of
Polish MO Finals
Subcontests
(6)
5
1
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property of tatrahedron with opposite edges equal
Prove that if in a tetrahedron
A
B
C
D
ABCD
A
BC
D
opposite edges are equal, i.e.
A
B
=
C
D
AB = CD
A
B
=
C
D
,
A
C
=
B
D
AC = BD
A
C
=
B
D
,
A
D
=
B
C
AD = BC
A
D
=
BC
, then the lines passing through the midpoints of opposite edges are mutually perpendicular and are the axes of symmetry of the tetrahedron.
6
1
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disk rolls insides a hook
Inside a hoop of radius
2
r
2r
2
r
a disk of radius
r
r
r
rolls on the hoop without slipping. What line is traced by a point arbitrarily chosen on the edge of the disk?
4
1
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\sqrt{x} - \sqrt{x- a} > 2
Find the values of
x
x
x
that satisfy the inequality
x
−
x
−
a
>
2
,
\sqrt{x} - \sqrt{x- a} > 2,
x
−
x
−
a
>
2
,
where
a
a
a
is a gicen poistive number.
3
1
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3 weights on a uniform circular disc
A uniform circular disc is suspended in a horizontal position on a string attached to its center
O
O
O
. At three different points
A
A
A
,
B
B
B
,
C
C
C
on the edge of the disc, weights
p
1
p_1
p
1
,
p
2
p_2
p
2
,
p
3
p_3
p
3
are placed, after which the disc remains in equilibrium. Calculate angles
A
O
B
AOB
A
OB
,
B
O
C
BOC
BOC
, and
C
O
A
COA
CO
A
.
2
1
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ctg A + \frac{cos B}{sin A cos C} = ctg B + \frac{cos A}{sin B cos C}.
What algebraic relationship holds between
A
A
A
,
B
B
B
, and
C
C
C
if
c
t
g
A
+
cos
B
sin
A
cos
C
=
c
t
g
B
+
cos
A
sin
B
cos
C
.
ctg A + \frac{\cos B}{\sin A \cos C} = ctg B + \frac{\cos A}{\sin B \cos C}.
c
t
g
A
+
sin
A
cos
C
cos
B
=
c
t
g
B
+
sin
B
cos
C
cos
A
.
1
1
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bicentric trapezoid
Prove that in an isosceles trapezoid circumscibed around a circle, the segments connecting the points of tangency of opposite sides with the circle pass through the point of intersection of the diagonals.