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Problems
Contests
National and Regional Contests
Hong Kong Contests
Hong Kong National Olympiad
2002 Hong kong National Olympiad
2002 Hong kong National Olympiad
Part of
Hong Kong National Olympiad
Subcontests
(4)
1
1
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Parallel tangents in two circles
Two circles meet at points
A
A
A
and
B
B
B
. A line through
B
B
B
intersects the first circle again at
K
K
K
and the second circle at
M
M
M
. A line parallel to
A
M
AM
A
M
is tangent to the first circle at
Q
Q
Q
. The line
A
Q
AQ
A
Q
intersects the second circle again at
R
R
R
.
(
a
)
(a)
(
a
)
Prove that the tangent to the second circle at
R
R
R
is parallel to
A
K
AK
A
K
.
(
b
)
(b)
(
b
)
Prove that these two tangents meet on
K
M
KM
K
M
.
4
1
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$p\equiv 1(\mod 4)$
Let
p
p
p
be a prime number such that
p
≡
1
(
m
o
d
4
)
p\equiv 1\pmod{4}
p
≡
1
(
mod
4
)
. Determine
∑
k
=
1
p
−
1
2
{
k
2
p
}
\sum_{k=1}^{\frac{p-1}{2}}\left \lbrace \frac{k^2}{p} \right \rbrace
∑
k
=
1
2
p
−
1
{
p
k
2
}
, where
{
x
}
=
x
−
[
x
]
\{x\}=x-[x]
{
x
}
=
x
−
[
x
]
.
3
1
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$a+b+c=3$
Let
a
≥
b
≥
c
≥
0
a\geq b\geq c\geq 0
a
≥
b
≥
c
≥
0
are real numbers such that
a
+
b
+
c
=
3
a+b+c=3
a
+
b
+
c
=
3
. Prove that
a
b
2
+
b
c
2
+
c
a
2
≤
27
8
ab^{2}+bc^{2}+ca^{2}\leq\frac{27}{8}
a
b
2
+
b
c
2
+
c
a
2
≤
8
27
and find cases of equality.
2
1
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conference there $n>2$ mathematicians
In conference there
n
>
2
n>2
n
>
2
mathematicians. Every two mathematicians communicate in one of the
n
n
n
offical languages of the conference. For any three different offical languages the exists three mathematicians who communicate with each other in these three languages. Find all
n
n
n
such that this is possible.