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Contests
International Contests
Silk Road
2008 Silk Road
2008 Silk Road
Part of
Silk Road
Subcontests
(4)
3
1
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let G be a graph with 2n vertexes and 2n(n-1) edges, we have red color
Let
G
G
G
be a graph with
2
n
2n
2
n
vertexes and 2n(n\minus{}1) edges.If we color some edge to red,then vertexes,which are connected by this edge,must be colored to red too. But not necessary that all edges from the red vertex are red. Prove that it is possible to color some vertexes and edges in
G
G
G
,such that all red vertexes has exactly
n
n
n
red edges.
2
1
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concurrent
In a triangle
A
B
C
ABC
A
BC
A
0
A_0
A
0
,
B
0
B_0
B
0
and
C
0
C_0
C
0
are the midpoints of the sides
B
C
BC
BC
,
C
A
CA
C
A
and
A
B
AB
A
B
.
A
1
A_1
A
1
,
B
1
B_1
B
1
,
C
1
C_1
C
1
are the midpoints of the broken lines
B
A
C
,
C
A
B
,
A
B
C
BAC,CAB,ABC
B
A
C
,
C
A
B
,
A
BC
.Show that
A
0
A
1
,
B
0
B
1
,
C
0
C
1
A_0A_1,B_0B_1,C_0C_1
A
0
A
1
,
B
0
B
1
,
C
0
C
1
are concurrent.
4
1
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Onto-Polynomials
Find all polynomials
P
∈
R
[
x
]
P\in\mathbb{R}[x]
P
∈
R
[
x
]
such that for all
r
∈
Q
r\in\mathbb{Q}
r
∈
Q
,there exist
d
∈
Q
d\in\mathbb{Q}
d
∈
Q
such that P(d)\equal{}r
1
1
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Silk road mathematical Olympiad _P1
Suppose
a
,
c
,
d
∈
N
a,c,d \in N
a
,
c
,
d
∈
N
and d|a^2b\plus{}c and d\geq a\plus{}c Prove that d\geq a\plus{}\sqrt[2b] {a}