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Problems
Contests
National and Regional Contests
Czech Republic Contests
District Round (Round II)
2009 District Round (Round II)
2009 District Round (Round II)
Part of
District Round (Round II)
Subcontests
(4)
4
1
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Czech republic,district round,2009,problem 4
in an acute triangle
A
B
C
ABC
A
BC
,
D
D
D
is a point on
B
C
BC
BC
,let
Q
Q
Q
be the intersection of
A
D
AD
A
D
and the median of
A
B
C
ABC
A
BC
from
C
C
C
,
P
P
P
is a point on
A
D
AD
A
D
,distinct from
Q
Q
Q
.the circumcircle of
C
P
D
CPD
CP
D
intersects
C
Q
CQ
CQ
at
C
C
C
and
K
K
K
.prove that the circumcircle of
A
K
P
AKP
A
K
P
passes through a fixed point differ from
A
A
A
.
3
1
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Czech republic,district round,2009,problem 3
A
,
B
,
C
A,B,C
A
,
B
,
C
are the three angles in a triangle such that
2
sin
B
sin
(
A
+
B
)
−
cos
A
=
1
2\sin B\sin (A+B)-\cos A=1
2
sin
B
sin
(
A
+
B
)
−
cos
A
=
1
,
2
sin
C
sin
(
B
+
C
)
−
cos
B
=
0
2\sin C\sin (B+C)-\cos B=0
2
sin
C
sin
(
B
+
C
)
−
cos
B
=
0
find the three angles.
2
1
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Czech republic,district round,2009,problem 2
in a right-angled triangle
A
B
C
ABC
A
BC
with
∠
C
=
90
\angle C=90
∠
C
=
90
,
a
,
b
,
c
a,b,c
a
,
b
,
c
are the corresponding sides.Circles
K
.
L
K.L
K
.
L
have their centers on
a
,
b
a,b
a
,
b
and are tangent to
b
,
c
b,c
b
,
c
;
a
,
c
a,c
a
,
c
respectively,with radii
r
,
t
r,t
r
,
t
.find the greatest real number
p
p
p
such that the inequality
1
r
+
1
t
≥
p
(
1
a
+
1
b
)
\frac{1}{r}+\frac{1}{t}\ge p(\frac{1}{a}+\frac{1}{b})
r
1
+
t
1
≥
p
(
a
1
+
b
1
)
always holds.
1
1
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Czech republic,district round,2009,problem 1
given a 4-digit number
(
a
b
c
d
)
10
(abcd)_{10}
(
ab
c
d
)
10
such that both
(
a
b
c
d
)
10
(abcd)_{10}
(
ab
c
d
)
10
and
(
d
c
b
a
)
10
(dcba)_{10}
(
d
c
ba
)
10
are multiples of
7
7
7
,having the same remainder modulo
37
37
37
.find
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
.