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Problems
Contests
National and Regional Contests
Poland Contests
Poland - Second Round
2000 Poland - Second Round
2000 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(6)
6
1
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Polynomial is a square of a polynomial
Polynomial
w
(
x
)
w(x)
w
(
x
)
of second degree with integer coefficients takes for integer arguments values, which are squares of integers. Prove that polynomial
w
(
x
)
w(x)
w
(
x
)
is a square of a polynomial.
5
1
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Decide whether function exists
Decide whether exists function
f
:
N
→
N
f: \mathbb{N} \rightarrow \mathbb{N}
f
:
N
→
N
, such that for each
n
∈
N
n \in \mathbb{N}
n
∈
N
is
f
(
f
(
n
)
)
=
2
n
f(f(n) )= 2n
f
(
f
(
n
))
=
2
n
.
4
1
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Determine all angle measures in triangle
Point
I
I
I
is incenter of triangle
A
B
C
ABC
A
BC
in which
A
B
≠
A
C
AB \neq AC
A
B
=
A
C
. Lines
B
I
BI
B
I
and
C
I
CI
C
I
intersect sides
A
C
AC
A
C
and
A
B
AB
A
B
in points
D
D
D
and
E
E
E
, respectively. Determine all measures of angle
B
A
C
BAC
B
A
C
, for which may be
D
I
=
E
I
DI = EI
D
I
=
E
I
.
3
1
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Admissible sets of chessboard fields
On fields of
n
×
n
n \times n
n
×
n
chessboard
n
2
n^2
n
2
different integers have been arranged, one in each field. In each column, field with biggest number was colored in red. Set of
n
n
n
fields of chessboard name admissible, if no two of that fields aren't in the same row and aren't in the same column. From all admissible sets, set with biggest sum of numbers in it's fields has been chosen. Prove that red field is in this set.
2
1
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Show inequality in triangle
Bisector of angle
B
A
C
BAC
B
A
C
of triangle
A
B
C
ABC
A
BC
intersects circumcircle of this triangle in point
D
≠
A
D \neq A
D
=
A
. Points
K
K
K
and
L
L
L
are orthogonal projections on line
A
D
AD
A
D
of points
B
B
B
and
C
C
C
, respectively. Prove that
A
D
≥
B
K
+
C
L
AD \ge BK + CL
A
D
≥
B
K
+
C
L
.
1
1
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Present rational number in the form of fraction
Decide, whether every positive rational number can present in the form
a
2
+
b
3
c
5
+
d
7
\frac{a^2 + b^3}{c^5 + d^7}
c
5
+
d
7
a
2
+
b
3
, where
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
are positive integers.