MathDB
Problems
Contests
National and Regional Contests
Hong Kong Contests
Hong Kong National Olympiad
2010 Hong kong National Olympiad
2010 Hong kong National Olympiad
Part of
Hong Kong National Olympiad
Subcontests
(3)
3
1
Hide problems
number theory equality
Let
n
n
n
be a positive integer. Let
a
a
a
be an integer such that
gcd
(
a
,
n
)
=
1
\gcd (a,n)=1
g
cd
(
a
,
n
)
=
1
. Prove that
a
ϕ
(
n
)
−
1
n
=
∑
i
∈
R
1
a
i
[
a
i
n
]
(
m
o
d
n
)
\frac{a^{\phi (n)}-1}{n}=\sum_{i\in R}\frac{1}{ai}\left[\frac{ai}{n}\right]\pmod{n}
n
a
ϕ
(
n
)
−
1
=
i
∈
R
∑
ai
1
[
n
ai
]
(
mod
n
)
where
R
R
R
is the reduced residue system of
n
n
n
with each element a positive integer at most
n
n
n
.
2
1
Hide problems
the number of sequences
Let
n
n
n
be a positive integer. Find the number of sequences
x
1
,
x
2
,
…
x
2
n
−
1
,
x
2
n
x_{1},x_{2},\ldots x_{2n-1},x_{2n}
x
1
,
x
2
,
…
x
2
n
−
1
,
x
2
n
, where
x
i
∈
{
−
1
,
1
}
x_{i}\in\{-1,1\}
x
i
∈
{
−
1
,
1
}
for each
i
i
i
, satisfying the following condition: for any integer
k
k
k
and
m
m
m
such that
1
≤
k
≤
m
≤
n
1\le k\le m\le n
1
≤
k
≤
m
≤
n
then the following inequality holds
∣
∑
i
=
2
k
−
1
2
m
x
i
∣
≤
2
\left|\sum_{i=2k-1}^{2m}x_{i}\right|\le\ 2
i
=
2
k
−
1
∑
2
m
x
i
≤
2
1
1
Hide problems
Centres of n-gons form an equilateral triangle
Let
A
B
C
ABC
A
BC
be an arbitrary triangle. A regular
n
n
n
-gon is constructed outward on the three sides of
△
A
B
C
\triangle ABC
△
A
BC
. Find all
n
n
n
such that the triangle formed by the three centres of the
n
n
n
-gons is equilateral.