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Problems
Contests
National and Regional Contests
Poland Contests
Polish MO Finals
1951 Polish MO Finals
1951 Polish MO Finals
Part of
Polish MO Finals
Subcontests
(6)
6
1
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construction of a point, similar triangles
Given a circle and a segment
M
N
MN
MN
. Find a point
C
C
C
on the circle such that the triangle
A
B
C
ABC
A
BC
, where
A
A
A
and
B
B
B
are the intersection points of the lines
M
C
MC
MC
and
N
C
NC
NC
with the circle, is similar to the triangle
M
N
C
MNC
MNC
.
5
1
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cyclic quad is rhombus
A quadrilateral
A
B
C
D
ABCD
A
BC
D
is inscribed in a circle. The lines
A
B
AB
A
B
and
C
D
CD
C
D
intersect at point
E
E
E
, and the lines
A
D
AD
A
D
and
B
C
BC
BC
intersect at point
F
F
F
. The bisector of the angle
A
E
C
AEC
A
EC
intersects the side
B
C
BC
BC
at the point
M
M
M
and the side
A
D
AD
A
D
at the point
N
N
N
; and the bisector of the angle
B
F
D
BFD
BF
D
intersects the side
A
B
AB
A
B
at the point
P
P
P
and the side
C
D
CD
C
D
at the point
Q
Q
Q
. Prove that the quadrilateral
M
P
N
Q
MPNQ
MPNQ
is a rhombus.
4
1
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x^3 - ax^2 + bx - c = 0
Determine the coefficients of 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
in such a way that the roots of this equation are the numbers
a
a
a
,
b
b
b
,
c
c
c
.
3
1
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ab (a + b) + bc (b + c) + ca (c + a) >= 6abc.
Prove that if
a
>
0
a > 0
a
>
0
,
b
>
0
b > 0
b
>
0
,
c
>
0
c > 0
c
>
0
, then the inequality holds
a
b
(
a
+
b
)
+
b
c
(
b
+
c
)
+
c
a
(
c
+
a
)
≥
6
a
b
c
.
ab (a + b) + bc (b + c) + ca (c + a) \geq 6abc.
ab
(
a
+
b
)
+
b
c
(
b
+
c
)
+
c
a
(
c
+
a
)
≥
6
ab
c
.
2
1
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30x0y03 divisible by $13
What digits should be placed instead of zeros in the third and fifth places in the number
3000003
3000003
3000003
to obtain a number divisible by
13
13
13
?
1
1
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beam and 2 parallel lines
A beam of length
a
a
a
is suspended horizontally with its ends on two parallel ropes equal
b
b
b
. We turn the beam through an angle
φ
\varphi
φ
around a vertical axis passing through the center of the beam. By how much will the beam rise?