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Problems
Contests
National and Regional Contests
Poland Contests
Polish MO Finals
1957 Polish MO Finals
1957 Polish MO Finals
Part of
Polish MO Finals
Subcontests
(6)
6
1
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distance of 2 lines in a cube
A cube is given with base
A
B
C
D
ABCD
A
BC
D
, where
A
B
=
a
AB = a
A
B
=
a
cm. Calculate the distance of the line
B
C
BC
BC
from the line passing through the point
A
A
A
and the center
S
S
S
of the face opposite the base.
5
1
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trisect segment
Given a line
m
m
m
and a segment
A
B
AB
A
B
parallel to it. Divide the segment
A
B
AB
A
B
into three equal parts using only a ruler, i.e. drawing only the lines.
4
1
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\sqrt{a^2 + b^2} \geq a + b - (2 - \sqrt{2}) \sqrt{ab}
Prove that if
a
≥
0
a \geq 0
a
≥
0
and
b
≥
0
b \geq 0
b
≥
0
, then
a
2
+
b
2
≥
a
+
b
−
(
2
−
2
)
a
b
.
\sqrt{a^2 + b^2} \geq a + b - (2 - \sqrt{2}) \sqrt{ab}.
a
2
+
b
2
≥
a
+
b
−
(
2
−
2
)
ab
.
3
1
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ax^2 + bx + c takes an integer value for every integer x
Prove that if the function
a
x
2
+
b
x
+
c
ax^2 + bx + c
a
x
2
+
b
x
+
c
takes an integer value for every integer value of the variable
x
x
x
, then
2
a
2a
2
a
,
a
+
b
a + b
a
+
b
,
c
c
c
are integers and vice versa.
2
1
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a^2 cos^2 A = b^2 cos^2 B + c^2 cos^2 C + 2bc cos B cos C cos 2A
Prove that between the sides
a
a
a
,
b
b
b
,
c
c
c
and the opposite angles
A
A
A
,
B
B
B
,
C
C
C
of a triangle there is a relationship
a
2
cos
2
A
=
b
2
cos
2
B
+
c
2
cos
2
C
+
2
b
c
cos
B
cos
C
cos
2
A
.
a^2 \cos^2 A = b^2 \cos^2 B + c^2 \cos^2 C + 2bc \cos B \cos C \cos 2A.
a
2
cos
2
A
=
b
2
cos
2
B
+
c
2
cos
2
C
+
2
b
c
cos
B
cos
C
cos
2
A
.
1
1
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orthogonal projection of segment MN on base = KL
Through the midpoint
S
S
S
of the segment
M
N
MN
MN
with endpoints lying on the legs of an isosceles triangle, a straight line is drawn parallel to the base of the triangle, intersecting its legs at points
K
K
K
and
L
L
L
. Prove that the orthogonal projection of the segment
M
N
MN
MN
onto the base of the triangle is equal to the segment
K
L
KL
K
L
.