MathDB
Problems
Contests
National and Regional Contests
China Contests
China Girls Math Olympiad
2019 China Girls Math Olympiad
2019 China Girls Math Olympiad
Part of
China Girls Math Olympiad
Subcontests
(8)
8
1
Hide problems
A tournament with 8 vertices
For a tournament with
8
8
8
vertices, if from any vertex it is impossible to follow a route to return to itself, we call the graph a good graph. Otherwise, we call it a bad graph. Prove that
(
1
)
(1)
(
1
)
there exists a tournament with
8
8
8
vertices such that after changing the orientation of any at most
7
7
7
edges of the tournament, the graph is always abad graph;
(
2
)
(2)
(
2
)
for any tournament with
8
8
8
vertices, one can change the orientation of at most
8
8
8
edges of the tournament to get a good graph. (A tournament is a complete graph with directed edges.)
7
1
Hide problems
Another cyclic quadrilateral in CGMO
Let
D
F
G
E
DFGE
D
FGE
be a cyclic quadrilateral. Line
D
F
DF
D
F
intersects
E
G
EG
EG
at
C
,
C,
C
,
and line
F
E
FE
FE
intersects
D
G
DG
D
G
at
H
.
H.
H
.
J
J
J
is the midpoint of
F
G
.
FG.
FG
.
The line
ℓ
\ell
ℓ
is the reflection of the line
D
E
DE
D
E
in
C
H
,
CH,
C
H
,
and it intersects line
G
F
GF
GF
at
I
.
I.
I
.
Prove that
C
,
J
,
H
,
I
C,J,H,I
C
,
J
,
H
,
I
are concyclic.
5
1
Hide problems
Chinese Girls Mathematical Olympiad 2019, Problem 5
Let
p
p
p
be a prime number such that
p
∣
(
2
2019
−
1
)
.
p\mid (2^{2019}-1) .
p
∣
(
2
2019
−
1
)
.
The sequence
a
1
,
a
2
,
.
.
.
,
a
n
a_1,a_2,...,a_n
a
1
,
a
2
,
...
,
a
n
satisfies the following conditions:
a
0
=
2
,
a
1
=
1
,
a
n
+
1
=
a
n
+
p
2
−
1
4
a
n
−
1
a_0=2, a_1=1 ,a_{n+1}=a_n+\frac{p^2-1}{4}a_{n-1}
a
0
=
2
,
a
1
=
1
,
a
n
+
1
=
a
n
+
4
p
2
−
1
a
n
−
1
(
n
≥
1
)
.
(n\geq 1).
(
n
≥
1
)
.
Prove that
p
∤
(
a
n
+
1
)
,
p\nmid (a_n+1),
p
∤
(
a
n
+
1
)
,
for any
n
≥
0.
n\geq 0.
n
≥
0.
6
1
Hide problems
Chinese Girls Mathematical Olympiad 2019, Problem 6
Let
0
≤
x
1
≤
x
2
≤
⋯
≤
x
n
≤
1
0\leq x_1\leq x_2\leq \cdots \leq x_n\leq 1
0
≤
x
1
≤
x
2
≤
⋯
≤
x
n
≤
1
(
n
≥
2
)
.
(n\geq 2).
(
n
≥
2
)
.
Prove that
x
1
x
2
⋯
x
n
n
+
(
1
−
x
1
)
(
1
−
x
2
)
⋯
(
1
−
x
n
)
n
≤
1
−
(
x
1
−
x
n
)
2
n
.
\sqrt[n]{x_1x_2 \cdots x_n}+ \sqrt[n]{(1-x_1)(1-x_2)\cdots (1-x_n)}\leq \sqrt[n]{1-(x_1- x_n)^2}.
n
x
1
x
2
⋯
x
n
+
n
(
1
−
x
1
)
(
1
−
x
2
)
⋯
(
1
−
x
n
)
≤
n
1
−
(
x
1
−
x
n
)
2
.
3
1
Hide problems
An operation to change a sequence into another one
For a sequence, one can perform the following operation: select three adjacent terms
a
,
b
,
c
,
a,b,c,
a
,
b
,
c
,
and change it into
b
,
c
,
a
.
b,c,a.
b
,
c
,
a
.
Determine all the possible positive integers
n
≥
3
,
n\geq 3,
n
≥
3
,
such that after finite number of operation, the sequence
1
,
2
,
⋯
,
n
1,2,\cdots, n
1
,
2
,
⋯
,
n
can be changed into
n
,
n
−
1
,
⋯
,
1
n,n-1,\cdots,1
n
,
n
−
1
,
⋯
,
1
finally.
1
1
Hide problems
A Cyclic Quadrilateral
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral with circumcircle
⊙
O
.
\odot O.
⊙
O
.
The lines tangent to
⊙
O
\odot O
⊙
O
at
A
,
B
A,B
A
,
B
intersect at
L
.
L.
L
.
M
M
M
is the midpoint of the segment
A
B
.
AB.
A
B
.
The line passing through
D
D
D
and parallel to
C
M
CM
CM
intersects
⊙
(
C
D
L
)
\odot (CDL)
⊙
(
C
D
L
)
at
F
.
F.
F
.
Line
C
F
CF
CF
intersects
D
M
DM
D
M
at
K
,
K,
K
,
and intersects
⊙
O
\odot O
⊙
O
at
E
E
E
(different from point
C
C
C
). Prove that
E
K
=
D
K
.
EK=DK.
E
K
=
DK
.
2
1
Hide problems
Chinese Girls Mathematical Olympiad 2019, Problem 2
Find integers
a
1
,
a
2
,
⋯
,
a
18
a_1,a_2,\cdots,a_{18}
a
1
,
a
2
,
⋯
,
a
18
, s.t.
a
1
=
1
,
a
2
=
2
,
a
18
=
2019
a_1=1,a_2=2,a_{18}=2019
a
1
=
1
,
a
2
=
2
,
a
18
=
2019
, and for all
3
≤
k
≤
18
3\le k\le 18
3
≤
k
≤
18
, there exists
1
≤
i
<
j
<
k
1\le i<j<k
1
≤
i
<
j
<
k
with
a
k
=
a
i
+
a
j
a_k=a_i+a_j
a
k
=
a
i
+
a
j
.
4
1
Hide problems
Chinese Girls Mathematical Olympiad 2019, Problem 4
Given parallelogram
O
A
B
C
OABC
O
A
BC
in the coodinate with
O
O
O
the origin and
A
,
B
,
C
A,B,C
A
,
B
,
C
be lattice points. Prove that for all lattice point
P
P
P
in the internal or boundary of
△
A
B
C
\triangle ABC
△
A
BC
, there exists lattice points
Q
,
R
Q,R
Q
,
R
(can be the same) in the internal or boundary of
△
O
A
C
\triangle OAC
△
O
A
C
with
O
P
→
=
O
Q
→
+
O
R
→
\overrightarrow{OP}=\overrightarrow{OQ}+\overrightarrow{OR}
OP
=
OQ
+
OR
.