MathDB
Problems
Contests
National and Regional Contests
China Contests
China Girls Math Olympiad
2014 China Girls Math Olympiad
2014 China Girls Math Olympiad
Part of
China Girls Math Olympiad
Subcontests
(8)
7
1
Hide problems
CGMO #7
Given a finite nonempty set
X
X
X
with real values, let
f
(
X
)
=
1
∣
X
∣
∑
a
∈
X
a
f(X) = \frac{1}{|X|} \displaystyle\sum\limits_{a\in X} a
f
(
X
)
=
∣
X
∣
1
a
∈
X
∑
a
, where
∣
X
∣
\left\lvert X \right\rvert
∣
X
∣
denotes the cardinality of
X
X
X
. For ordered pairs of sets
(
A
,
B
)
(A,B)
(
A
,
B
)
such that
A
∪
B
=
{
1
,
2
,
…
,
100
}
A\cup B = \{1, 2, \dots , 100\}
A
∪
B
=
{
1
,
2
,
…
,
100
}
and
A
∩
B
=
∅
A\cap B = \emptyset
A
∩
B
=
∅
where
1
≤
∣
A
∣
≤
98
1\leq |A| \leq 98
1
≤
∣
A
∣
≤
98
, select some
p
∈
B
p\in B
p
∈
B
, and let
A
p
=
A
∪
{
p
}
A_{p} = A\cup \{p\}
A
p
=
A
∪
{
p
}
and
B
p
=
B
−
{
p
}
.
B_{p} = B - \{p\}.
B
p
=
B
−
{
p
}
.
Over all such
(
A
,
B
)
(A,B)
(
A
,
B
)
and
p
∈
B
p\in B
p
∈
B
determine the maximum possible value of
(
f
(
A
p
)
−
f
(
A
)
)
(
f
(
B
p
)
−
f
(
B
)
)
.
(f(A_{p})-f(A))(f(B_{p})-f(B)).
(
f
(
A
p
)
−
f
(
A
))
(
f
(
B
p
)
−
f
(
B
))
.
8
1
Hide problems
CGMO #8
Let
n
n
n
be a positive integer, and set
S
S
S
be the set of all integers in
{
1
,
2
,
…
,
n
}
\{1,2,\dots,n\}
{
1
,
2
,
…
,
n
}
which are relatively prime to
n
n
n
. Set
S
1
=
S
∩
(
0
,
n
3
]
S_1 = S \cap \left(0, \frac n3 \right]
S
1
=
S
∩
(
0
,
3
n
]
,
S
2
=
S
∩
(
n
3
,
2
n
3
]
S_2 = S \cap \left( \frac n3, \frac {2n}3 \right]
S
2
=
S
∩
(
3
n
,
3
2
n
]
,
S
3
=
S
∩
(
2
n
3
,
n
]
S_3 = S \cap \left( \frac{2n}{3}, n \right]
S
3
=
S
∩
(
3
2
n
,
n
]
. If the cardinality of
S
S
S
is a multiple of
3
3
3
, prove that
S
1
S_1
S
1
,
S
2
S_2
S
2
,
S
3
S_3
S
3
have the same cardinality.
6
1
Hide problems
CGMO #6
In acute triangle
A
B
C
ABC
A
BC
,
A
B
>
A
C
AB > AC
A
B
>
A
C
.
D
D
D
and
E
E
E
are the midpoints of
A
B
AB
A
B
,
A
C
AC
A
C
respectively. The circumcircle of
A
D
E
ADE
A
D
E
intersects the circumcircle of
B
C
E
BCE
BCE
again at
P
P
P
. The circumcircle of
A
D
E
ADE
A
D
E
intersects the circumcircle
B
C
D
BCD
BC
D
again at
Q
Q
Q
. Prove that
A
P
=
A
Q
AP = AQ
A
P
=
A
Q
.
4
1
Hide problems
CGMO #4
For an integer
m
≥
4
,
m\geq 4,
m
≥
4
,
let
T
m
T_{m}
T
m
denote the number of sequences
a
1
,
…
,
a
m
a_{1},\dots,a_{m}
a
1
,
…
,
a
m
such that the following conditions hold: (1) For all
i
=
1
,
2
,
…
,
m
i=1,2,\dots,m
i
=
1
,
2
,
…
,
m
we have
a
i
∈
{
1
,
2
,
3
,
4
}
a_{i}\in \{1,2,3,4\}
a
i
∈
{
1
,
2
,
3
,
4
}
(2)
a
1
=
a
m
=
1
a_{1} = a_{m} = 1
a
1
=
a
m
=
1
and
a
2
≠
1
a_{2}\neq 1
a
2
=
1
(3) For all
i
=
3
,
4
⋯
,
m
,
a
i
≠
a
i
−
1
,
a
i
≠
a
i
−
2
.
i=3,4\cdots, m, a_{i}\neq a_{i-1}, a_{i}\neq a_{i-2}.
i
=
3
,
4
⋯
,
m
,
a
i
=
a
i
−
1
,
a
i
=
a
i
−
2
.
Prove that there exists a geometric sequence of positive integers
{
g
n
}
\{g_{n}\}
{
g
n
}
such that for
n
≥
4
n\geq 4
n
≥
4
we have that
g
n
−
2
g
n
<
T
n
<
g
n
+
2
g
n
.
g_{n} - 2\sqrt{g_{n}} < T_{n} < g_{n} + 2\sqrt{g_{n}}.
g
n
−
2
g
n
<
T
n
<
g
n
+
2
g
n
.
1
1
Hide problems
CGMO #1
In the figure of http://www.artofproblemsolving.com/Forum/download/file.php?id=50643&mode=view
⊙
O
1
\odot O_1
⊙
O
1
and
⊙
O
2
\odot O_2
⊙
O
2
intersect at two points
A
A
A
,
B
B
B
. The extension of
O
1
A
O_1A
O
1
A
meets
⊙
O
2
\odot O_2
⊙
O
2
at
C
C
C
, and the extension of
O
2
A
O_2A
O
2
A
meets
⊙
O
1
\odot O_1
⊙
O
1
at
D
D
D
, and through
B
B
B
draw
B
E
∥
O
2
A
BE \parallel O_2A
BE
∥
O
2
A
intersecting
⊙
O
1
\odot O_1
⊙
O
1
again at
E
E
E
. If
D
E
∥
O
1
A
DE \parallel O_1A
D
E
∥
O
1
A
, prove that
D
C
⊥
C
O
2
DC \perp CO_2
D
C
⊥
C
O
2
.
3
1
Hide problems
Chinese Girls Mathematical Olympiad 2014, Problem 3
There are
n
n
n
students; each student knows exactly
d
d
d
girl students and
d
d
d
boy students ("knowing" is a symmetric relation). Find all pairs
(
n
,
d
)
(n,d)
(
n
,
d
)
of integers .
2
1
Hide problems
Chinese Girls Mathematical Olympiad 2014, Problem 2
Let
x
1
,
x
2
,
…
,
x
n
x_1,x_2,\ldots,x_n
x
1
,
x
2
,
…
,
x
n
be real numbers, where
n
≥
2
n\ge 2
n
≥
2
is a given integer, and let
⌊
x
1
⌋
,
⌊
x
2
⌋
,
…
,
⌊
x
n
⌋
\lfloor{x_1}\rfloor,\lfloor{x_2}\rfloor,\ldots,\lfloor{x_n}\rfloor
⌊
x
1
⌋
,
⌊
x
2
⌋
,
…
,
⌊
x
n
⌋
be a permutation of
1
,
2
,
…
,
n
1,2,\ldots,n
1
,
2
,
…
,
n
. Find the maximum and minimum of
∑
i
=
1
n
−
1
⌊
x
i
+
1
−
x
i
⌋
\sum\limits_{i=1}^{n-1}\lfloor{x_{i+1}-x_i}\rfloor
i
=
1
∑
n
−
1
⌊
x
i
+
1
−
x
i
⌋
(here
⌊
x
⌋
\lfloor x\rfloor
⌊
x
⌋
is the largest integer not greater than
x
x
x
).
5
1
Hide problems
Chinese Girls Mathematical Olympiad 2014, Problem 5
Let
a
a
a
be a positive integer, but not a perfect square;
r
r
r
is a real root of the equation
x
3
−
2
a
x
+
1
=
0
x^3-2ax+1=0
x
3
−
2
a
x
+
1
=
0
. Prove that
r
+
a
r+\sqrt{a}
r
+
a
is an irrational number.