MathDB
Problems
Contests
National and Regional Contests
China Contests
(China) National High School Mathematics League
2014 China Second Round Olympiad
2014 China Second Round Olympiad
Part of
(China) National High School Mathematics League
Subcontests
(4)
4
1
Hide problems
2014 China Second Round Olympiad Second Part Problem 4
Let
x
1
,
x
2
,
…
,
x
2014
x_1,x_2,\dots,x_{2014}
x
1
,
x
2
,
…
,
x
2014
be integers among which no two are congurent modulo
2014
2014
2014
. Let
y
1
,
y
2
,
…
,
y
2014
y_1,y_2,\dots,y_{2014}
y
1
,
y
2
,
…
,
y
2014
be integers among which no two are congurent modulo
2014
2014
2014
. Prove that one can rearrange
y
1
,
y
2
,
…
,
y
2014
y_1,y_2,\dots,y_{2014}
y
1
,
y
2
,
…
,
y
2014
to
z
1
,
z
2
,
…
,
z
2014
z_1,z_2,\dots,z_{2014}
z
1
,
z
2
,
…
,
z
2014
, so that among
x
1
+
z
1
,
x
2
+
z
2
,
…
,
x
2014
+
z
2014
x_1+z_1,x_2+z_2,\dots,x_{2014}+z_{2014}
x
1
+
z
1
,
x
2
+
z
2
,
…
,
x
2014
+
z
2014
no two are congurent modulo
4028
4028
4028
.
3
1
Hide problems
2014 China Second Round Olympiad Second Part Problem 3
Let
S
=
{
1
,
2
,
3
,
⋯
,
100
}
S=\{1,2,3,\cdots,100\}
S
=
{
1
,
2
,
3
,
⋯
,
100
}
. Find the maximum value of integer
k
k
k
, such that there exist
k
k
k
different nonempty subsets of
S
S
S
satisfying the condition: for any two of the
k
k
k
subsets, if their intersection is nonemply, then the minimal element of their intersection is not equal to the maximal element of either of the two subsets.
2
1
Hide problems
2014 China Second Round Olympiad Second Part Problem 2
Let
A
B
C
ABC
A
BC
be an acute triangle such that
∠
B
A
C
≠
6
0
∘
\angle BAC \neq 60^\circ
∠
B
A
C
=
6
0
∘
. Let
D
,
E
D,E
D
,
E
be points such that
B
D
,
C
E
BD,CE
B
D
,
CE
are tangent to the circumcircle of
A
B
C
ABC
A
BC
and
B
D
=
C
E
=
B
C
BD=CE=BC
B
D
=
CE
=
BC
(
A
A
A
is on one side of line
B
C
BC
BC
and
D
,
E
D,E
D
,
E
are on the other side). Let
F
,
G
F,G
F
,
G
be intersections of line
D
E
DE
D
E
and lines
A
B
,
A
C
AB,AC
A
B
,
A
C
. Let
M
M
M
be intersection of
C
F
CF
CF
and
B
D
BD
B
D
, and
N
N
N
be intersection of
C
E
CE
CE
and
B
G
BG
BG
. Prove that
A
M
=
A
N
AM=AN
A
M
=
A
N
.
1
1
Hide problems
China Second Round Olympiad 2014 Second Part Problem 1
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be real numbers such that
a
+
b
+
c
=
1
a+b+c=1
a
+
b
+
c
=
1
and
a
b
c
>
0
abc>0
ab
c
>
0
. Prove that
b
c
+
c
a
+
a
b
<
a
b
c
2
+
1
4
.
bc+ca+ab<\frac{\sqrt{abc}}{2}+\frac{1}{4}.
b
c
+
c
a
+
ab
<
2
ab
c
+
4
1
.