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Contests
National and Regional Contests
China Contests
China Girls Math Olympiad
2013 China Girls Math Olympiad
2013 China Girls Math Olympiad
Part of
China Girls Math Olympiad
Subcontests
(8)
8
1
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2013 China girls' Mathematical Olympiad problem 8
Let
n
n
n
(
≥
4
\ge 4
≥
4
) be an even integer. We label
n
n
n
pairwise distinct real numbers arbitrarily on the
n
n
n
vertices of a regular
n
n
n
-gon, and label the
n
n
n
sides clockwise as
e
1
,
e
2
,
…
,
e
n
e_1, e_2, \ldots, e_n
e
1
,
e
2
,
…
,
e
n
. A side is called positive if the numbers on both endpoints are increasing in clockwise direction. An unordered pair of distinct sides
{
e
i
,
e
j
}
\left\{ e_i,e_j \right\}
{
e
i
,
e
j
}
is called alternating if it satisfies both conditions:(i)
2
∣
(
i
+
j
)
2 \mid (i+j)
2
∣
(
i
+
j
)
; and(ii) if one rearranges the four numbers on the vertices of these two sides
e
i
e_i
e
i
and
e
j
e_j
e
j
in increasing order
a
<
b
<
c
<
d
a < b < c < d
a
<
b
<
c
<
d
, then
a
a
a
and
c
c
c
are the numbers on the two endpoints of one of sides
e
i
e_i
e
i
or
e
j
e_j
e
j
.Prove that the number of alternating pairs of sides and the number of positive sides are of different parity.
7
1
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2013 China girls' Mathematical Olympiad problem 7
As shown in the figure,
⊙
O
1
\odot O_1
⊙
O
1
and
⊙
O
2
\odot O_2
⊙
O
2
touches each other externally at a point
T
T
T
, quadrilateral
A
B
C
D
ABCD
A
BC
D
is inscribed in
⊙
O
1
\odot O_1
⊙
O
1
, and the lines
D
A
DA
D
A
,
C
B
CB
CB
are tangent to
⊙
O
2
\odot O_2
⊙
O
2
at points
E
E
E
and
F
F
F
respectively. Line
B
N
BN
BN
bisects
∠
A
B
F
\angle ABF
∠
A
BF
and meets segment
E
F
EF
EF
at
N
N
N
. Line
F
T
FT
FT
meets the arc
A
T
^
\widehat{AT}
A
T
(not passing through the point
B
B
B
) at another point
M
M
M
different from
A
A
A
. Prove that
M
M
M
is the circumcenter of
△
B
C
N
\triangle BCN
△
BCN
.
2
1
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2013 China girls' Mathematical Olympiad problem 2
As shown in the figure below,
A
B
C
D
ABCD
A
BC
D
is a trapezoid,
A
B
∥
C
D
AB \parallel CD
A
B
∥
C
D
. The sides
D
A
DA
D
A
,
A
B
AB
A
B
,
B
C
BC
BC
are tangent to
⊙
O
1
\odot O_1
⊙
O
1
and
A
B
AB
A
B
touches
⊙
O
1
\odot O_1
⊙
O
1
at
P
P
P
. The sides
B
C
BC
BC
,
C
D
CD
C
D
,
D
A
DA
D
A
are tangent to
⊙
O
2
\odot O_2
⊙
O
2
, and
C
D
CD
C
D
touches
⊙
O
2
\odot O_2
⊙
O
2
at
Q
Q
Q
. Prove that the lines
A
C
AC
A
C
,
B
D
BD
B
D
,
P
Q
PQ
PQ
meet at the same point.[asy] size(200); defaultpen(linewidth(0.8)+fontsize(10pt)); pair A=origin,B=(1,-7),C=(30,-15),D=(26,6); pair bisA=bisectorpoint(B,A,D),bisB=bisectorpoint(A,B,C),bisC=bisectorpoint(B,C,D),bisD=bisectorpoint(C,D,A); path bA=A--(bisA+100*(bisA-A)),bB=B--(bisB+100*(bisB-B)),bC=C--(bisC+100*(bisC-C)),bD=D--(bisD+100*(bisD-D)); pair O1=intersectionpoint(bA,bB),O2=intersectionpoint(bC,bD); dot(O1^^O2,linewidth(2)); pair h1=foot(O1,A,B),h2=foot(O2,C,D); real r1=abs(O1-h1),r2=abs(O2-h2); draw(circle(O1,r1)^^circle(O2,r2)); draw(A--B--C--D--cycle); draw(A--C^^B--D^^h1--h2); label("
A
A
A
",A,NW); label("
B
B
B
",B,SW); label("
C
C
C
",C,dir(350)); label("
D
D
D
",D,dir(350)); label("
P
P
P
",h1,dir(200)); label("
Q
Q
Q
",h2,dir(350)); label("
O
1
O_1
O
1
",O1,dir(150)); label("
O
2
O_2
O
2
",O2,dir(300)); [/asy]
3
1
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Chinese Girls Mathematical Olympiad 2013, Problem 3
In a group of
m
m
m
girls and
n
n
n
boys, any two persons either know each other or do not know each other. For any two boys and any two girls, there are at least one boy and one girl among them,who do not know each other. Prove that the number of unordered pairs of (boy, girl) who know each other does not exceed
m
+
n
(
n
−
1
)
2
m+\frac{n(n-1)}{2}
m
+
2
n
(
n
−
1
)
.
5
1
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Chinese Girls Mathematical Olympiad 2013, Problem 5
For any given positive numbers
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\ldots,a_n
a
1
,
a
2
,
…
,
a
n
, prove that there exist positive numbers
x
1
,
x
2
,
…
,
x
n
x_1,x_2,\ldots,x_n
x
1
,
x
2
,
…
,
x
n
satisfying
∑
i
=
1
n
x
i
=
1
\sum_{i=1}^n x_i=1
∑
i
=
1
n
x
i
=
1
, such that for any positive numbers
y
1
,
y
2
,
…
,
y
n
y_1,y_2,\ldots,y_n
y
1
,
y
2
,
…
,
y
n
with
∑
i
=
1
n
y
i
=
1
\sum_{i=1}^n y_i=1
∑
i
=
1
n
y
i
=
1
, the inequality
∑
i
=
1
n
a
i
x
i
x
i
+
y
i
≥
1
2
∑
i
=
1
n
a
i
\sum_{i=1}^n \frac{a_ix_i}{x_i+y_i}\ge \frac{1}{2}\sum_{i=1}^n a_i
∑
i
=
1
n
x
i
+
y
i
a
i
x
i
≥
2
1
∑
i
=
1
n
a
i
holds.
6
1
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Chinese Girls Mathematical Olympiad 2013, Problem 6
Let
S
S
S
be a subset of
{
0
,
1
,
2
,
…
,
98
}
\{0,1,2,\ldots,98 \}
{
0
,
1
,
2
,
…
,
98
}
with exactly
m
≥
3
m\geq 3
m
≥
3
(distinct) elements, such that for any
x
,
y
∈
S
x,y\in S
x
,
y
∈
S
there exists
z
∈
S
z\in S
z
∈
S
satisfying
x
+
y
≡
2
z
(
m
o
d
99
)
x+y \equiv 2z \pmod{99}
x
+
y
≡
2
z
(
mod
99
)
. Determine all possible values of
m
m
m
.
4
1
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Chinese Girls Mathematical Olympiad 2013, Problem 4
Find the number of polynomials
f
(
x
)
=
a
x
3
+
b
x
f(x)=ax^3+bx
f
(
x
)
=
a
x
3
+
b
x
satisfying both following conditions: (i)
a
,
b
∈
{
1
,
2
,
…
,
2013
}
a,b\in\{1,2,\ldots,2013\}
a
,
b
∈
{
1
,
2
,
…
,
2013
}
; (ii) the difference between any two of
f
(
1
)
,
f
(
2
)
,
…
,
f
(
2013
)
f(1),f(2),\ldots,f(2013)
f
(
1
)
,
f
(
2
)
,
…
,
f
(
2013
)
is not a multiple of
2013
2013
2013
.
1
1
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Chinese Girls Mathematical Olympiad 2013, Problem 1
Let
A
A
A
be the closed region bounded by the following three lines in the
x
y
xy
x
y
plane:
x
=
1
,
y
=
0
x=1, y=0
x
=
1
,
y
=
0
and
y
=
t
(
2
x
−
t
)
y=t(2x-t)
y
=
t
(
2
x
−
t
)
, where
0
<
t
<
1
0<t<1
0
<
t
<
1
. Prove that the area of any triangle inside the region
A
A
A
, with two vertices
P
(
t
,
t
2
)
P(t,t^2)
P
(
t
,
t
2
)
and
Q
(
1
,
0
)
Q(1,0)
Q
(
1
,
0
)
, does not exceed
1
4
.
\frac{1}{4}.
4
1
.