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International Contests
Silk Road
2010 Silk Road
2010 Silk Road
Part of
Silk Road
Subcontests
(4)
1
1
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convex ABCD, <ADB +<ACB = <CAB + <DBA =30^o, AD=BC, right triangle
In a convex quadrilateral it is known
A
B
C
D
ABCD
A
BC
D
that
∠
A
D
B
+
∠
A
C
B
=
∠
C
A
B
+
∠
D
B
A
=
3
0
∘
\angle ADB + \angle ACB = \angle CAB + \angle DBA = 30^{\circ}
∠
A
D
B
+
∠
A
CB
=
∠
C
A
B
+
∠
D
B
A
=
3
0
∘
and
A
D
=
B
C
AD = BC
A
D
=
BC
. Prove that from the lengths
D
B
DB
D
B
,
C
A
CA
C
A
and
D
C
DC
D
C
, you can make a right triangle.
3
1
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SRMC 2010 P3
For positive real numbers
a
,
b
,
c
,
d
,
a, b, c, d,
a
,
b
,
c
,
d
,
satisfying the following conditions:
a
(
c
2
−
1
)
=
b
(
b
2
+
c
2
)
a(c^2 - 1)=b(b^2+c^2)
a
(
c
2
−
1
)
=
b
(
b
2
+
c
2
)
and
d
≤
1
d \leq 1
d
≤
1
, prove that :
d
(
a
1
−
d
2
+
b
2
1
+
d
2
)
≤
(
a
+
b
)
c
2
d(a \sqrt{1-d^2} + b^2 \sqrt{1+d^2}) \leq \frac{(a+b)c}{2}
d
(
a
1
−
d
2
+
b
2
1
+
d
2
)
≤
2
(
a
+
b
)
c
2
1
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SRMC 2010 P2
Let
N
=
2010
!
+
1
N = 2010!+1
N
=
2010
!
+
1
. Prove that a)
N
N
N
is not divisible by
4021
4021
4021
; b)
N
N
N
is not divisible by
2027
,
2029
,
2039
2027,2029,2039
2027
,
2029
,
2039
; c)
N
N
N
has a prime divisor greater than
2050
2050
2050
.
4
1
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Country with two capitals (SRMC 2010)
In country there are two capitals (
A
A
A
and
B
B
B
) and finite number of towns. Some towns (or town with one of capital) connected with roads (one-way). (between every two towns or capital and town there are arbitrary number of roads) such that exist at least one way from
A
A
A
to
B
B
B
. Given, that any two ways from
A
A
A
to
B
B
B
have at least one common road. Prove, that exist one road, such that all ways from
A
A
A
to
B
B
B
pass through this road.