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National and Regional Contests
Poland Contests
Polish MO Finals
2004 Polish MO Finals
2004 Polish MO Finals
Part of
Polish MO Finals
Subcontests
(6)
6
1
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Successive terms of a sequence which are divisible by m
An integer
m
>
1
m > 1
m
>
1
is given. The infinite sequence
(
x
n
)
n
≥
0
(x_n)_{n\ge 0}
(
x
n
)
n
≥
0
is defined by x_i\equal{}2^i for
i
<
m
i<m
i
<
m
and x_i\equal{}x_{i\minus{}1}\plus{}x_{i\minus{}2}\plus{}\cdots \plus{}x_{i\minus{}m} for
i
≥
m
i\ge m
i
≥
m
. Find the greatest natural number
k
k
k
such that there exist
k
k
k
successive terms of this sequence which are divisible by
m
m
m
.
2
1
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polynomial with integer coefficients
Let
P
P
P
be a polynomial with integer coefficients such that there are two distinct integers at which
P
P
P
takes coprime values. Show that there exists an infinite set of integers, such that the values
P
P
P
takes at them are pairwise coprime.
4
1
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inequality of three real numbers
Let real numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
. Prove that \sqrt{2(a^2\plus{}b^2)}\plus{}\sqrt{2(b^2\plus{}c^2)}\plus{}\sqrt{2(c^2\plus{}a^2)}\ge \sqrt{3(a\plus{}b)^2\plus{}3(b\plus{}c)^2\plus{}3(c\plus{}a)^2}.
3
1
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draw-triple in a tournament
On a tournament with
n
≥
3
n \ge 3
n
≥
3
participants, every two participants played exactly one match and there were no draws. A three-element set of participants is called a draw-triple if they can be enumerated so that the first defeated the second, the second defeated the third, and the third defeated the first. Determine the largest possible number of draw-triples on such a tournament.
5
1
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the angle between any two lines is the same
Find the greatest possible number of lines in space that all pass through a single point and the angle between any two of them is the same.
1
1
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the points D,E, F lie on a line
A point
D
D
D
is taken on the side
A
B
AB
A
B
of a triangle
A
B
C
ABC
A
BC
. Two circles passing through
D
D
D
and touching
A
C
AC
A
C
and
B
C
BC
BC
at
A
A
A
and
B
B
B
respectively intersect again at point
E
E
E
. Let
F
F
F
be the point symmetric to
C
C
C
with respect to the perpendicular bisector of
A
B
AB
A
B
. Prove that the points
D
,
E
,
F
D,E,F
D
,
E
,
F
lie on a line.