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Problems
Contests
National and Regional Contests
Poland Contests
Polish MO Finals
1960 Polish MO Finals
1960 Polish MO Finals
Part of
Polish MO Finals
Subcontests
(6)
6
1
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min path, rectangle
On the perimeter of a rectangle, point
M
M
M
is chosen. Find the shortest path whose beginning and end are point
M
M
M
and which has a point in common with each side of the rectangle.
5
1
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sums of 4-digit numbers
From the digits
1
1
1
,
2
2
2
,
3
3
3
,
4
4
4
,
5
5
5
,
6
6
6
,
7
7
7
,
8
8
8
,
9
9
9
all possible four-digit numbers with different digits are formed. Find the sum of these numbers.
4
1
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( a/4)^4 = (b/3)^3 if x^4 + ax + b = 0 has two equal roots,
Prove that if the equation
x
4
+
a
x
+
b
=
0
x^4 + ax + b = 0
x
4
+
a
x
+
b
=
0
has two equal roots, then
(
a
4
)
4
=
(
b
3
)
3
.
\left( \frac{a}{4} \right)^4 =\left( \frac{b}{3} \right)^3.
(
4
a
)
4
=
(
3
b
)
3
.
3
1
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parallel in cyclic hexagon
On the circle 6 distinct points
A
A
A
,
B
B
B
,
C
C
C
,
D
D
D
,
E
E
E
,
F
F
F
are chosen in such a way that
A
B
AB
A
B
is parallel to
D
E
DE
D
E
, and
D
C
DC
D
C
is parallel to
A
F
AF
A
F
. Prove that
B
C
BC
BC
is parallel to
E
F
EF
EF
2
1
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tg^2 a + tg^2 \b + tg^2 c=12.$
A plane is drawn through the height of a regular tetrahedron, which intersects the planes of the lateral faces along
3
3
3
lines that form angles
α
\alpha
α
,
β
\beta
β
,
γ
\gamma
γ
with the plane of the tetrahedron's base. Prove that
t
g
2
α
+
t
g
2
β
+
t
g
2
γ
=
12.
tg^2 \alpha + tg^2 \beta + tg^2 \gamma =12.
t
g
2
α
+
t
g
2
β
+
t
g
2
γ
=
12.
1
1
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2^n > n^2
Prove that if
n
n
n
is an integer greater than
4
4
4
, then
2
n
2^n
2
n
is greater than
n
2
n^2
n
2
.