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Contests
National and Regional Contests
China Contests
China National Olympiad
2010 China National Olympiad
2010 China National Olympiad
Part of
China National Olympiad
Subcontests
(3)
2
2
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China Mathematics Olympiads (National Round) 2010 Problem 2
Let
k
k
k
be an integer
≥
3
\geq 3
≥
3
. Sequence
{
a
n
}
\{a_n\}
{
a
n
}
satisfies that
a
k
=
2
k
a_k = 2k
a
k
=
2
k
and for all
n
>
k
n > k
n
>
k
, we have
a
n
=
{
a
n
−
1
+
1
if
(
a
n
−
1
,
n
)
=
1
2
n
if
(
a
n
−
1
,
n
)
>
1
a_n = \begin{cases} a_{n-1}+1 & \text{if } (a_{n-1},n) = 1 \\ 2n & \text{if } (a_{n-1},n) > 1 \end{cases}
a
n
=
{
a
n
−
1
+
1
2
n
if
(
a
n
−
1
,
n
)
=
1
if
(
a
n
−
1
,
n
)
>
1
Prove that there are infinitely many primes in the sequence
{
a
n
−
a
n
−
1
}
\{a_n - a_{n-1}\}
{
a
n
−
a
n
−
1
}
.
China Mathematics Olympiads (National Round) 2010 Problem 5
There is a deck of cards placed at every points
A
1
,
A
2
,
…
,
A
n
A_1, A_2, \ldots , A_n
A
1
,
A
2
,
…
,
A
n
and
O
O
O
, where
n
≥
3
n \geq 3
n
≥
3
. We can do one of the following two operations at each step:
1
)
1)
1
)
If there are more than 2 cards at some points
A
i
A_i
A
i
, we can withdraw three cards from that deck and place one each at
A
i
−
1
,
A
i
+
1
A_{i-1}, A_{i+1}
A
i
−
1
,
A
i
+
1
and
O
O
O
. (Here
A
0
=
A
n
A_0=A_n
A
0
=
A
n
and
A
n
+
1
=
A
1
A_{n+1}=A_1
A
n
+
1
=
A
1
);
2
)
2)
2
)
If there are more than or equal to
n
n
n
cards at point
O
O
O
, we can withdraw
n
n
n
cards from that deck and place one each at
A
1
,
A
2
,
…
,
A
n
A_1, A_2, \ldots , A_n
A
1
,
A
2
,
…
,
A
n
. Show that if the total number of cards is more than or equal to
n
2
+
3
n
+
1
n^2+3n+1
n
2
+
3
n
+
1
, we can make the number of cards at every points more than or equal to
n
+
1
n+1
n
+
1
after finitely many steps.
1
2
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China Mathematics Olympiads (National Round) 2010 Problem 1
Two circles
Γ
1
\Gamma_1
Γ
1
and
Γ
2
\Gamma_2
Γ
2
meet at
A
A
A
and
B
B
B
. A line through
B
B
B
meets
Γ
1
\Gamma_1
Γ
1
and
Γ
2
\Gamma_2
Γ
2
again at
C
C
C
and
D
D
D
repsectively. Another line through
B
B
B
meets
Γ
1
\Gamma_1
Γ
1
and
Γ
2
\Gamma_2
Γ
2
again at
E
E
E
and
F
F
F
repsectively. Line
C
F
CF
CF
meets
Γ
1
\Gamma_1
Γ
1
and
Γ
2
\Gamma_2
Γ
2
again at
P
P
P
and
Q
Q
Q
respectively.
M
M
M
and
N
N
N
are midpoints of arc
P
B
PB
PB
and arc
Q
B
QB
QB
repsectively. Show that if
C
D
=
E
F
CD = EF
C
D
=
EF
, then
C
,
F
,
M
,
N
C,F,M,N
C
,
F
,
M
,
N
are concyclic.
Subset of N such that inequality holds
Let
m
,
n
≥
1
m,n\ge 1
m
,
n
≥
1
and
a
1
<
a
2
<
…
<
a
n
a_1 < a_2 < \ldots < a_n
a
1
<
a
2
<
…
<
a
n
be integers. Prove that there exists a subset
T
T
T
of
N
\mathbb{N}
N
such that
∣
T
∣
≤
1
+
a
n
−
a
1
2
n
+
1
|T| \leq 1+ \frac{a_n-a_1}{2n+1}
∣
T
∣
≤
1
+
2
n
+
1
a
n
−
a
1
and for every
i
∈
{
1
,
2
,
…
,
m
}
i \in \{1,2,\ldots , m\}
i
∈
{
1
,
2
,
…
,
m
}
, there exists
t
∈
T
t \in T
t
∈
T
and
s
∈
[
−
n
,
n
]
s \in [-n,n]
s
∈
[
−
n
,
n
]
, such that
a
i
=
t
+
s
a_i=t+s
a
i
=
t
+
s
.
3
2
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China Mathematics Olympiads (National Round) 2010 Problem 3
Given complex numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
, we have that
∣
a
z
2
+
b
z
+
c
∣
≤
1
|az^2 + bz +c| \leq 1
∣
a
z
2
+
b
z
+
c
∣
≤
1
holds true for any complex number
z
,
∣
z
∣
≤
1
z, |z| \leq 1
z
,
∣
z
∣
≤
1
. Find the maximum value of
∣
b
c
∣
|bc|
∣
b
c
∣
.
b_i=ka_i
Suppose
a
1
,
a
2
,
a
3
,
b
1
,
b
2
,
b
3
a_1,a_2,a_3,b_1,b_2,b_3
a
1
,
a
2
,
a
3
,
b
1
,
b
2
,
b
3
are distinct positive integers such that (n \plus{} 1)a_1^n \plus{} na_2^n \plus{} (n \minus{} 1)a_3^n|(n \plus{} 1)b_1^n \plus{} nb_2^n \plus{} (n \minus{} 1)b_3^n holds for all positive integers
n
n
n
. Prove that there exists
k
∈
N
k\in N
k
∈
N
such that b_i \equal{} ka_i for i \equal{} 1,2,3.