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Contests
National and Regional Contests
China Contests
China National Olympiad
2011 China National Olympiad
2011 China National Olympiad
Part of
China National Olympiad
Subcontests
(3)
3
2
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union of the subsets 2011 China national Olmypaid 3
Let
A
A
A
be a set consist of finite real numbers,
A
1
,
A
2
,
⋯
,
A
n
A_1,A_2,\cdots,A_n
A
1
,
A
2
,
⋯
,
A
n
be nonempty sets of
A
A
A
, such that (a) The sum of the elements of
A
A
A
is
0
,
0,
0
,
(b) For all
x
i
∈
A
i
(
i
=
1
,
2
,
⋯
,
n
)
x_i \in A_i(i=1,2,\cdots,n)
x
i
∈
A
i
(
i
=
1
,
2
,
⋯
,
n
)
,we have
x
1
+
x
2
+
⋯
+
x
n
>
0
x_1+x_2+\cdots+x_n>0
x
1
+
x
2
+
⋯
+
x
n
>
0
.Prove that there exist
1
≤
k
≤
n
,
1\le k\le n,
1
≤
k
≤
n
,
and
1
≤
i
1
<
i
2
<
⋯
<
i
k
≤
n
1\le i_1<i_2<\cdots<i_k\le n
1
≤
i
1
<
i
2
<
⋯
<
i
k
≤
n
, such that
∣
A
i
1
⋃
A
i
2
⋃
⋯
⋃
A
i
k
∣
<
k
n
∣
A
∣
.
|A_{i_1}\bigcup A_{i_2} \bigcup \cdots \bigcup A_{i_k}|<\frac{k}{n}|A|.
∣
A
i
1
⋃
A
i
2
⋃
⋯
⋃
A
i
k
∣
<
n
k
∣
A
∣.
Where
∣
X
∣
|X|
∣
X
∣
denote the numbers of the elements in set
X
X
X
.
infinite couple of a,b 2011 China national Olmypiad 6
Let
m
,
n
m,n
m
,
n
be positive integer numbers. Prove that there exist infinite many couples of positive integer nubmers
(
a
,
b
)
(a,b)
(
a
,
b
)
such that a+b| am^a+bn^b , \gcd(a,b)=1.
2
2
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Prove the incenter on line 2011 China national Olmypaid 2
On the circumcircle of the acute triangle
A
B
C
ABC
A
BC
,
D
D
D
is the midpoint of
B
C
⌢
\stackrel{\frown}{BC}
BC
⌢
. Let
X
X
X
be a point on
B
D
⌢
\stackrel{\frown}{BD}
B
D
⌢
,
E
E
E
the midpoint of
A
X
⌢
\stackrel{\frown}{AX}
A
X
⌢
, and let
S
S
S
lie on
A
C
⌢
\stackrel{\frown}{AC}
A
C
⌢
. The lines
S
D
SD
S
D
and
B
C
BC
BC
have intersection
R
R
R
, and the lines
S
E
SE
SE
and
A
X
AX
A
X
have intersection
T
T
T
. If
R
T
∥
D
E
RT \parallel DE
RT
∥
D
E
, prove that the incenter of the triangle
A
B
C
ABC
A
BC
is on the line
R
T
.
RT.
RT
.
sums 2011 China national Olympiad 5
Let
a
i
,
b
i
,
i
=
1
,
⋯
,
n
a_i,b_i,i=1,\cdots,n
a
i
,
b
i
,
i
=
1
,
⋯
,
n
are nonnegitive numbers,and
n
≥
4
n\ge 4
n
≥
4
,such that
a
1
+
a
2
+
⋯
+
a
n
=
b
1
+
b
2
+
⋯
+
b
n
>
0
a_1+a_2+\cdots+a_n=b_1+b_2+\cdots+b_n>0
a
1
+
a
2
+
⋯
+
a
n
=
b
1
+
b
2
+
⋯
+
b
n
>
0
.Find the maximum of
∑
i
=
1
n
a
i
(
a
i
+
b
i
)
∑
i
=
1
n
b
i
(
a
i
+
b
i
)
\frac{\sum_{i=1}^n a_i(a_i+b_i)}{\sum_{i=1}^n b_i(a_i+b_i)}
∑
i
=
1
n
b
i
(
a
i
+
b
i
)
∑
i
=
1
n
a
i
(
a
i
+
b
i
)
1
2
Hide problems
Cyclic inequality China 2011 national olympiad 1
Let
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\ldots,a_n
a
1
,
a
2
,
…
,
a
n
are real numbers, prove that;
∑
i
=
1
n
a
i
2
−
∑
i
=
1
n
a
i
a
i
+
1
≤
⌊
n
2
⌋
(
M
−
m
)
2
.
\sum_{i=1}^na_i^2-\sum_{i=1}^n a_i a_{i+1} \le \left\lfloor \frac{n}{2}\right\rfloor(M-m)^2.
i
=
1
∑
n
a
i
2
−
i
=
1
∑
n
a
i
a
i
+
1
≤
⌊
2
n
⌋
(
M
−
m
)
2
.
where
a
n
+
1
=
a
1
,
M
=
max
1
≤
i
≤
n
a
i
,
m
=
min
1
≤
i
≤
n
a
i
a_{n+1}=a_1,M=\max_{1\le i\le n} a_i,m=\min_{1\le i\le n} a_i
a
n
+
1
=
a
1
,
M
=
max
1
≤
i
≤
n
a
i
,
m
=
min
1
≤
i
≤
n
a
i
.
Find the minmum 2011 China national Olmypaid 4
Let
n
n
n
be an given positive integer, the set
S
=
{
1
,
2
,
⋯
,
n
}
S=\{1,2,\cdots,n\}
S
=
{
1
,
2
,
⋯
,
n
}
.For any nonempty set
A
A
A
and
B
B
B
, find the minimum of
∣
A
Δ
S
∣
+
∣
B
Δ
S
∣
+
∣
C
Δ
S
∣
,
|A\Delta S|+|B\Delta S|+|C\Delta S|,
∣
A
Δ
S
∣
+
∣
B
Δ
S
∣
+
∣
C
Δ
S
∣
,
where
C
=
{
a
+
b
∣
a
∈
A
,
b
∈
B
}
,
X
Δ
Y
=
X
∪
Y
−
X
∩
Y
.
C=\{a+b|a\in A,b\in B\}, X\Delta Y=X\cup Y-X\cap Y.
C
=
{
a
+
b
∣
a
∈
A
,
b
∈
B
}
,
X
Δ
Y
=
X
∪
Y
−
X
∩
Y
.