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Contests
National and Regional Contests
Poland Contests
Polish MO Finals
2006 Polish MO Finals
2006 Polish MO Finals
Part of
Polish MO Finals
Subcontests
(3)
3
1
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Convex hexagon and midpoints
Let
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
be a convex hexagon satisfying
A
C
=
D
F
AC=DF
A
C
=
D
F
,
C
E
=
F
B
CE=FB
CE
=
FB
and
E
A
=
B
D
EA=BD
E
A
=
B
D
. Prove that the lines connecting the midpoints of opposite sides of the hexagon
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
intersect in one point.
2
2
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exponents + perfect power
Find all positive integers
k
k
k
for which number
3
k
+
5
k
3^k+5^k
3
k
+
5
k
is a power of some integer with exponent greater than
1
1
1
.
Tetrahedron, insphere and centroids
Tetrahedron
A
B
C
D
ABCD
A
BC
D
in which
A
B
=
C
D
AB=CD
A
B
=
C
D
is given. Sphere inscribed in it is tangent to faces
A
B
C
ABC
A
BC
and
A
B
D
ABD
A
B
D
respectively in
K
K
K
and
L
L
L
. Prove that if points
K
K
K
and
L
L
L
are centroids of faces
A
B
C
ABC
A
BC
and
A
B
D
ABD
A
B
D
then tetrahedron
A
B
C
D
ABCD
A
BC
D
is regular.
1
2
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System of equations - standard
Solve in reals: \begin{eqnarray*}a^2=b^3+c^3 \\ b^2=c^3+d^3 \\ c^2=d^3+e^3 \\ d^2=e^3+a^3 \\ e^2=a^3+b^3 \end{eqnarray*}
operation on triplet
Given a triplet we perform on it the following operation. We choose two numbers among them and change them into their sum and product, left number stays unchanged. Can we, starting from triplet
(
3
,
4
,
5
)
(3,4,5)
(
3
,
4
,
5
)
and performing above operation, obtain again a triplet of numbers which are lengths of right triangle?