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Problems
Contests
National and Regional Contests
Hong Kong Contests
Hong Kong National Olympiad
2009 Hong kong National Olympiad
2009 Hong kong National Olympiad
Part of
Hong Kong National Olympiad
Subcontests
(4)
4
1
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find all pairs (m,n)
find all pairs of non-negative integer pairs
(
m
,
n
)
(m,n)
(
m
,
n
)
,satisfies
10
7
56
(
m
2
−
1
)
+
2
m
+
3
=
(
11
3
114
n
)
107^{56}(m^{2}-1)+2m+3=\binom{113^{114}}{n}
10
7
56
(
m
2
−
1
)
+
2
m
+
3
=
(
n
11
3
114
)
3
1
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2009 HongKong mathematical Olympiad
A
B
C
ABC
A
BC
is a right triangle with
∠
C
=
90
\angle C=90
∠
C
=
90
,
C
D
CD
C
D
is perpendicular to
A
B
AB
A
B
,and
D
D
D
is the foot,
ω
\omega
ω
is the circumcircle of triangle
B
C
D
BCD
BC
D
,
ω
1
\omega_{1}
ω
1
is a circle inside triangle
A
C
D
ACD
A
C
D
,tangent to
A
D
AD
A
D
and
A
C
AC
A
C
at
M
M
M
and
N
N
N
respectively,and
ω
1
\omega_{1}
ω
1
is also tangent to
ω
\omega
ω
.prove that: (1)
B
D
∗
C
N
+
B
C
∗
D
M
=
C
D
∗
B
M
BD*CN+BC*DM=CD*BM
B
D
∗
CN
+
BC
∗
D
M
=
C
D
∗
BM
(2)
B
M
=
B
C
BM=BC
BM
=
BC
2
1
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a graph of three colours
there are
n
n
n
points on the plane,any two vertex are connected by an edge of red,yellow or green,and any triangle with vertex in the graph contains exactly
2
2
2
colours.prove that
n
<
13
n<13
n
<
13
1
1
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sequence
let
a
n
{a_{n}}
a
n
be a sequence of integers,
a
1
a_{1}
a
1
is odd,and for any positive integer
n
n
n
,we have
n
(
a
n
+
1
−
a
n
+
3
)
=
a
n
+
1
+
a
n
+
3
n(a_{n+1}-a_{n}+3)=a_{n+1}+a_{n}+3
n
(
a
n
+
1
−
a
n
+
3
)
=
a
n
+
1
+
a
n
+
3
,in addition,we have
2010
2010
2010
divides
a
2009
a_{2009}
a
2009
find the smallest
n
≥
2
n\ge\ 2
n
≥
2
,so that
2010
2010
2010
divides
a
n
a_{n}
a
n