MathDB
Problems
Contests
National and Regional Contests
China Contests
China Girls Math Olympiad
2011 China Girls Math Olympiad
2011 China Girls Math Olympiad
Part of
China Girls Math Olympiad
Subcontests
(8)
7
1
Hide problems
Each box has one ball in it
There are
n
n
n
boxes
B
1
,
B
2
,
…
,
B
n
{B_1},{B_2},\ldots,{B_n}
B
1
,
B
2
,
…
,
B
n
from left to right, and there are
n
n
n
balls in these boxes. If there is at least
1
1
1
ball in
B
1
{B_1}
B
1
, we can move one to
B
2
{B_2}
B
2
. If there is at least
1
1
1
ball in
B
n
{B_n}
B
n
, we can move one to
B
n
−
1
{B_{n - 1}}
B
n
−
1
. If there are at least
2
2
2
balls in
B
k
{B_k}
B
k
,
2
≤
k
≤
n
−
1
2 \leq k \leq n - 1
2
≤
k
≤
n
−
1
we can move one to
B
k
−
1
{B_{k - 1}}
B
k
−
1
, and one to
B
k
+
1
{B_{k + 1}}
B
k
+
1
. Prove that, for any arrangement of the
n
n
n
balls, we can achieve that each box has one ball in it.
4
1
Hide problems
Tennis tournament
A tennis tournament has
n
>
2
n>2
n
>
2
players and any two players play one game against each other (ties are not allowed). After the game these players can be arranged in a circle, such that for any three players
A
,
B
,
C
A,B,C
A
,
B
,
C
, if
A
,
B
A,B
A
,
B
are adjacent on the circle, then at least one of
A
,
B
A,B
A
,
B
won against
C
C
C
. Find all possible values for
n
n
n
.
5
1
Hide problems
Find the smallest $\lambda$
A real number
α
≥
0
\alpha \geq 0
α
≥
0
is given. Find the smallest
λ
=
λ
(
α
)
>
0
\lambda = \lambda (\alpha ) > 0
λ
=
λ
(
α
)
>
0
, such that for any complex numbers
z
1
,
z
2
{z_1},{z_2}
z
1
,
z
2
and
0
≤
x
≤
1
0 \leq x \leq 1
0
≤
x
≤
1
, if
∣
z
1
∣
≤
α
∣
z
1
−
z
2
∣
\left| {{z_1}} \right| \leq \alpha \left| {{z_1} - {z_2}} \right|
∣
z
1
∣
≤
α
∣
z
1
−
z
2
∣
, then
∣
z
1
−
x
z
2
∣
≤
λ
∣
z
1
−
z
2
∣
\left| {{z_1} - x{z_2}} \right| \leq \lambda \left| {{z_1} - {z_2}} \right|
∣
z
1
−
x
z
2
∣
≤
λ
∣
z
1
−
z
2
∣
.
1
1
Hide problems
Equation has 2011 positive integer solutions (x,y)
Find all positive integers
n
n
n
such that the equation
1
x
+
1
y
=
1
n
\frac{1}{x} + \frac{1}{y} = \frac{1}{n}
x
1
+
y
1
=
n
1
has exactly
2011
2011
2011
positive integer solutions
(
x
,
y
)
(x,y)
(
x
,
y
)
where
x
≤
y
x \leq y
x
≤
y
.
6
1
Hide problems
Can m^20 + 11^n be a square?
Do there exist positive integers
m
,
n
m,n
m
,
n
, such that
m
20
+
1
1
n
m^{20}+11^n
m
20
+
1
1
n
is a square number?
2
1
Hide problems
PQ is perpendicular to BC
The diagonals
A
C
,
B
D
AC,BD
A
C
,
B
D
of the quadrilateral
A
B
C
D
ABCD
A
BC
D
intersect at
E
E
E
. Let
M
,
N
M,N
M
,
N
be the midpoints of
A
B
,
C
D
AB,CD
A
B
,
C
D
respectively. Let the perpendicular bisectors of the segments
A
B
,
C
D
AB,CD
A
B
,
C
D
meet at
F
F
F
. Suppose that
E
F
EF
EF
meets
B
C
,
A
D
BC,AD
BC
,
A
D
at
P
,
Q
P,Q
P
,
Q
respectively. If
M
F
⋅
C
D
=
N
F
⋅
A
B
MF\cdot CD=NF\cdot AB
MF
⋅
C
D
=
NF
⋅
A
B
and
D
Q
⋅
B
P
=
A
Q
⋅
C
P
DQ\cdot BP=AQ\cdot CP
D
Q
⋅
BP
=
A
Q
⋅
CP
, prove that
P
Q
⊥
B
C
PQ\perp BC
PQ
⊥
BC
.
8
1
Hide problems
Midpoint of FG lies on MN
The
A
A
A
-excircle
(
O
)
(O)
(
O
)
of
△
A
B
C
\triangle ABC
△
A
BC
touches
B
C
BC
BC
at
M
M
M
. The points
D
,
E
D,E
D
,
E
lie on the sides
A
B
,
A
C
AB,AC
A
B
,
A
C
respectively such that
D
E
∥
B
C
DE\parallel BC
D
E
∥
BC
. The incircle
(
O
1
)
(O_1)
(
O
1
)
of
△
A
D
E
\triangle ADE
△
A
D
E
touches
D
E
DE
D
E
at
N
N
N
. If
B
O
1
∩
D
O
=
F
BO_1\cap DO=F
B
O
1
∩
D
O
=
F
and
C
O
1
∩
E
O
=
G
CO_1\cap EO=G
C
O
1
∩
EO
=
G
, prove that the midpoint of
F
G
FG
FG
lies on
M
N
MN
MN
.
3
1
Hide problems
Not easy one
The positive reals
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
satisfy
a
b
c
d
=
1
abcd=1
ab
c
d
=
1
. Prove that
1
a
+
1
b
+
1
c
+
1
d
+
9
a
+
b
+
c
+
d
⩾
25
4
\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} + \frac{9}{{a + b + c + d}} \geqslant \frac{{25}}{4}
a
1
+
b
1
+
c
1
+
d
1
+
a
+
b
+
c
+
d
9
⩾
4
25
.