MathDB
Problems
Contests
National and Regional Contests
China Contests
China Girls Math Olympiad
2012 China Girls Math Olympiad
2012 China Girls Math Olympiad
Part of
China Girls Math Olympiad
Subcontests
(8)
8
1
Hide problems
CGMO8: How many k such that 2012 divides nCr(2012,k)
Find the number of integers
k
k
k
in the set
{
0
,
1
,
2
,
…
,
2012
}
\{0, 1, 2, \dots, 2012\}
{
0
,
1
,
2
,
…
,
2012
}
such that
(
2012
k
)
\binom{2012}{k}
(
k
2012
)
is a multiple of
2012
2012
2012
.
7
1
Hide problems
CGMO7: Nondecreasing Sequence
Let
{
a
n
}
\{a_n\}
{
a
n
}
be a sequence of nondecreasing positive integers such that
r
a
r
=
k
+
1
\textstyle\frac{r}{a_r} = k+1
a
r
r
=
k
+
1
for some positive integers
k
k
k
and
r
r
r
. Prove that there exists a positive integer
s
s
s
such that
s
a
s
=
k
\textstyle\frac{s}{a_s} = k
a
s
s
=
k
.
6
1
Hide problems
CGMO6: Airline companies and cities
There are
n
n
n
cities,
2
2
2
airline companies in a country. Between any two cities, there is exactly one
2
2
2
-way flight connecting them which is operated by one of the two companies. A female mathematician plans a travel route, so that it starts and ends at the same city, passes through at least two other cities, and each city in the route is visited once. She finds out that wherever she starts and whatever route she chooses, she must take flights of both companies. Find the maximum value of
n
n
n
.
5
1
Hide problems
CGMO5: Carlos Shine's Fact 5
As shown in the figure below, the in-circle of
A
B
C
ABC
A
BC
is tangent to sides
A
B
AB
A
B
and
A
C
AC
A
C
at
D
D
D
and
E
E
E
respectively, and
O
O
O
is the circumcenter of
B
C
I
BCI
BC
I
. Prove that
∠
O
D
B
=
∠
O
E
C
\angle ODB = \angle OEC
∠
O
D
B
=
∠
OEC
. [asy]import graph; size(5.55cm); pathpen=linewidth(0.7); pointpen=black; pen fp=fontsize(10); pointfontpen=fp; real xmin=-5.76,xmax=4.8,ymin=-3.69,ymax=3.71; pen zzttqq=rgb(0.6,0.2,0), wwwwqq=rgb(0.4,0.4,0), qqwuqq=rgb(0,0.39,0); pair A=(-2,2.5), B=(-3,-1.5), C=(2,-1.5), I=(-1.27,-0.15), D=(-2.58,0.18), O=(-0.5,-2.92); D(A--B--C--cycle,zzttqq); D(arc(D,0.25,-104.04,-56.12)--(-2.58,0.18)--cycle,qqwuqq); D(arc((-0.31,0.81),0.25,-92.92,-45)--(-0.31,0.81)--cycle,qqwuqq); D(A--B,zzttqq); D(B--C,zzttqq); D(C--A,zzttqq); D(CR(I,1.35),linewidth(1.2)+dotted+wwwwqq); D(CR(O,2.87),linetype("2 2")+blue); D(D--O); D((-0.31,0.81)--O); D(A); D(B); D(C); D(I); D(D); D((-0.31,0.81)); D(O); MP( "A", A, dir(110)); MP("B", B, dir(140)); D("C", C, dir(20)); D("D", D, dir(150)); D("E", (-0.31, 0.81), dir(60)); D("O", O, dir(290)); D("I", I, dir(100)); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy]
4
1
Hide problems
CGMO4: Stones in a regular 13-gon
There is a stone at each vertex of a given regular
13
13
13
-gon, and the color of each stone is black or white. Prove that we may exchange the position of two stones such that the coloring of these stones are symmetric with respect to some symmetric axis of the
13
13
13
-gon.
3
1
Hide problems
CGMO3: Nontrivial gcd of a^n+b^n+1 for all n>=1
Find all pairs
(
a
,
b
)
(a,b)
(
a
,
b
)
of integers satisfying: there exists an integer
d
≥
2
d \ge 2
d
≥
2
such that
a
n
+
b
n
+
1
a^n + b^n +1
a
n
+
b
n
+
1
is divisible by
d
d
d
for all positive integers
n
n
n
.
2
1
Hide problems
CGMO2: Mutually Tangent Circles
Circles
Q
1
Q_1
Q
1
and
Q
2
Q_2
Q
2
are tangent to each other externally at
T
T
T
. Points
A
A
A
and
E
E
E
are on
Q
1
Q_1
Q
1
, lines
A
B
AB
A
B
and
D
E
DE
D
E
are tangent to
Q
2
Q_2
Q
2
at
B
B
B
and
D
D
D
, respectively, lines
A
E
AE
A
E
and
B
D
BD
B
D
meet at point
P
P
P
. Prove that (1)
A
B
A
T
=
E
D
E
T
\frac{AB}{AT}=\frac{ED}{ET}
A
T
A
B
=
ET
E
D
; (2)
∠
A
T
P
+
∠
E
T
P
=
18
0
∘
\angle ATP + \angle ETP = 180^{\circ}
∠
A
TP
+
∠
ETP
=
18
0
∘
. [asy]import graph; size(5.97cm); real lsf=0.5; pathpen=linewidth(0.7); pointpen=black; pen fp=fontsize(10); pointfontpen=fp; real xmin=-6,xmax=5.94,ymin=-3.19,ymax=3.43; pair Q_1=(-2.5,-0.5), T=(-1.5,-0.5), Q_2=(0.5,-0.5), A=(-2.09,0.41), B=(-0.42,1.28), D=(-0.2,-2.37), P=(-0.52,2.96); D(CR(Q_1,1)); D(CR(Q_2,2)); D(A--B); D((-3.13,-1.27)--D); D(P--(-3.13,-1.27)); D(P--D); D(T--(-3.13,-1.27)); D(T--A); D(T--P); D(Q_1); MP("Q_1",(-2.46,-0.44),NE*lsf); D(T); MP("T",(-1.46,-0.44),NE*lsf); D(Q_2); MP("Q_2",(0.54,-0.44),NE*lsf); D(A); MP("A",(-2.22,0.58),NE*lsf); D(B); MP("B",(-0.35,1.45),NE*lsf); D((-3.13,-1.27)); MP("E",(-3.52,-1.62),NE*lsf); D(D); MP("D",(-0.17,-2.31),NE*lsf); D(P); MP("P",(-0.47,3.02),NE*lsf); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy]
1
1
Hide problems
Fractional Inequality
Let
a
1
,
a
2
,
…
,
a
n
a_1, a_2,\ldots, a_n
a
1
,
a
2
,
…
,
a
n
be non-negative real numbers. Prove that
1
1
+
a
1
+
a
1
(
1
+
a
1
)
(
1
+
a
2
)
+
a
1
a
2
(
1
+
a
1
)
(
1
+
a
2
)
(
1
+
a
3
)
+
\frac{1}{1+ a_1}+\frac{ a_1}{(1+ a_1)(1+ a_2)}+\frac{ a_1 a_2}{(1+ a_1)(1+ a_2)(1+ a_3)}+
1
+
a
1
1
+
(
1
+
a
1
)
(
1
+
a
2
)
a
1
+
(
1
+
a
1
)
(
1
+
a
2
)
(
1
+
a
3
)
a
1
a
2
+
⋯
+
a
1
a
2
⋯
a
n
−
1
(
1
+
a
1
)
(
1
+
a
2
)
⋯
(
1
+
a
n
)
≤
1.
\cdots+\frac{ a_1 a_2\cdots a_{n-1}}{(1+ a_1)(1+ a_2)\cdots (1+ a_n)} \le 1.
⋯
+
(
1
+
a
1
)
(
1
+
a
2
)
⋯
(
1
+
a
n
)
a
1
a
2
⋯
a
n
−
1
≤
1.