MathDB
Problems
Contests
National and Regional Contests
Hong Kong Contests
Hong Kong National Olympiad
2008 Hong kong National Olympiad
2008 Hong kong National Olympiad
Part of
Hong Kong National Olympiad
Subcontests
(4)
1
1
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Iterated function of a polynomial
Let f(x) \equal{} c_m x^m \plus{} c_{m\minus{}1} x^{m\minus{}1} \plus{}...\plus{} c_1 x \plus{} c_0, where each
c
i
c_i
c
i
is a non-zero integer. Define a sequence
{
a
n
}
\{ a_n \}
{
a
n
}
by a_1 \equal{} 0 and a_{n\plus{}1} \equal{} f(a_n) for all positive integers
n
n
n
. (a) Let
i
i
i
and
j
j
j
be positive integers with
i
<
j
i<j
i
<
j
. Show that a_{j\plus{}1} \minus{} a_j is a multiple of a_{i\plus{}1} \minus{} a_i. (b) Show that
a
2008
≠
0
a_{2008} \neq 0
a
2008
=
0
4
1
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Minimum number of intersection among 2008 congruent circles
There are 2008 congruent circles on a plane such that no two are tangent to each other and each circle intersects at least three other circles. Let
N
N
N
be the total number of intersection points of these circles. Determine the smallest possible values of
N
N
N
.
3
1
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Prove H is the orthocentre from given condition
Δ
A
B
C
\Delta ABC
Δ
A
BC
is a triangle such that
A
B
≠
A
C
AB \neq AC
A
B
=
A
C
. The incircle of
Δ
A
B
C
\Delta ABC
Δ
A
BC
touches
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
at
D
,
E
,
F
D, E, F
D
,
E
,
F
respectively.
H
H
H
is a point on the segment
E
F
EF
EF
such that
D
H
⊥
E
F
DH \bot EF
DH
⊥
EF
. Suppose
A
H
⊥
B
C
AH \bot BC
A
H
⊥
BC
, prove that
H
H
H
is the orthocentre of
Δ
A
B
C
\Delta ABC
Δ
A
BC
. Remark: the original question has missed the condition
A
B
≠
A
C
AB \neq AC
A
B
=
A
C
2
1
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Euler's totient function and sum of divisor function
Let
n
>
4
n>4
n
>
4
be a positive integer such that
n
n
n
is composite (not a prime) and divides \varphi (n) \sigma (n) \plus{}1, where
φ
(
n
)
\varphi (n)
φ
(
n
)
is the Euler's totient function of
n
n
n
and
σ
(
n
)
\sigma (n)
σ
(
n
)
is the sum of the positive divisors of
n
n
n
. Prove that
n
n
n
has at least three distinct prime factors.