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Problems
Contests
National and Regional Contests
Czech Republic Contests
District Round (Round II)
2008 District Round (Round II)
2008 District Round (Round II)
Part of
District Round (Round II)
Subcontests
(4)
4
1
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Czech republic,district round,2008,problem 4
A semicircle has diameter
A
B
AB
A
B
and center
S
S
S
,with a point
M
M
M
on the circumference.
U
,
V
U,V
U
,
V
are the incircles of sectors
A
S
M
ASM
A
SM
and
B
S
M
BSM
BSM
.Prove that circles
U
,
V
U,V
U
,
V
can be seperated by a line perpendicular to
A
B
AB
A
B
.
3
1
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Czech republic,district round,2008,problem 3
For
n
>
2
n>2
n
>
2
, an
n
×
n
n\times n
n
×
n
grid of squares is coloured black and white like a chessboard, with its upper left corner coloured black. Then we can recolour some of the white squares black in the following way: choose a
2
×
3
2\times 3
2
×
3
(or
3
×
2
3\times 2
3
×
2
) rectangle which has exactly
3
3
3
white squares and then colour all these
3
3
3
white squares black. Find all
n
n
n
such that after a series of such operations all squares will be black.
2
1
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Czech republic,district round,2008,problem 2
Two circles
U
,
V
U,V
U
,
V
have distinct radii,tangent to each other externally at
T
T
T
.
A
,
B
A,B
A
,
B
are points on
U
,
V
U,V
U
,
V
respectively,both distinct from
T
T
T
,such that
∠
A
T
B
=
90
\angle ATB=90
∠
A
TB
=
90
. (1)Prove that line
A
B
AB
A
B
passes through a fixed point; (2)Find the locus of the midpoint of
A
B
AB
A
B
.
1
1
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Czech republic,district round,2008,problem 1
Let
n
n
n
be an integer greater than
1
1
1
.Find all pairs of integers
(
s
,
t
)
(s,t)
(
s
,
t
)
such that equations:
x
n
+
s
x
=
2007
x^n+sx=2007
x
n
+
s
x
=
2007
and
x
n
+
t
x
=
2008
x^n+tx=2008
x
n
+
t
x
=
2008
have at least one common real root.