MathDB
Problems
Contests
National and Regional Contests
China Contests
China Girls Math Olympiad
2022 China Girls Math Olympiad
2022 China Girls Math Olympiad
Part of
China Girls Math Olympiad
Subcontests
(8)
8
1
Hide problems
Certain Condition When The Equality Holds
Let
x
1
,
x
2
,
…
,
x
11
x_1, x_2, \ldots, x_{11}
x
1
,
x
2
,
…
,
x
11
be nonnegative reals such that their sum is
1
1
1
. For
i
=
1
,
2
,
…
,
11
i = 1,2, \ldots, 11
i
=
1
,
2
,
…
,
11
, let
y
i
=
{
x
i
+
x
i
+
1
,
i
odd
,
x
i
+
x
i
+
1
+
x
i
+
2
,
i
even
,
y_i = \begin{cases} x_{i} + x_{i + 1}, & i \, \, \textup{odd} ,\\ x_{i} + x_{i + 1} + x_{i + 2}, & i \, \, \textup{even} ,\end{cases}
y
i
=
{
x
i
+
x
i
+
1
,
x
i
+
x
i
+
1
+
x
i
+
2
,
i
odd
,
i
even
,
where
x
12
=
x
1
x_{12} = x_{1}
x
12
=
x
1
. And let
F
(
x
1
,
x
2
,
…
,
x
11
)
=
y
1
y
2
…
y
11
F (x_1, x_2, \ldots, x_{11}) = y_1 y_2 \ldots y_{11}
F
(
x
1
,
x
2
,
…
,
x
11
)
=
y
1
y
2
…
y
11
. Prove that
x
6
<
x
8
x_6 < x_8
x
6
<
x
8
when
F
F
F
achieves its maximum.
7
1
Hide problems
Polygon 3-coloring covered by triangles with different color vertices
Let
n
⩾
3
n \geqslant 3
n
⩾
3
be integer. Given convex
n
−
n-
n
−
polygon
P
\mathcal{P}
P
. A
3
−
3-
3
−
coloring of the vertices of
P
\mathcal{P}
P
is called nice such that every interior point of
P
\mathcal{P}
P
is inside or on the bound of a triangle formed by polygon vertices with pairwise distinct colors. Determine the number of different nice colorings. ([I]Two colorings are different as long as they differ at some vertices. )
6
1
Hide problems
Sets covering all integers mod n
Find all integers
n
n
n
satisfying the following property. There exist nonempty finite integer sets
A
A
A
and
B
B
B
such that for any integer
m
m
m
, exactly one of these three statements below is true: (a) There is
a
∈
A
a \in A
a
∈
A
such that
m
≡
a
(
m
o
d
n
)
m \equiv a \pmod n
m
≡
a
(
mod
n
)
, (b) There is
b
∈
B
b \in B
b
∈
B
such that
m
≡
b
(
m
o
d
n
)
m \equiv b \pmod n
m
≡
b
(
mod
n
)
, and (c) There are
a
∈
A
a \in A
a
∈
A
and
b
∈
B
b \in B
b
∈
B
such that
m
≡
a
+
b
(
m
o
d
n
)
m \equiv a + b \pmod n
m
≡
a
+
b
(
mod
n
)
.
5
1
Hide problems
Easy Geometry
Two points
K
K
K
and
L
L
L
are chosen inside triangle
A
B
C
ABC
A
BC
and a point
D
D
D
is chosen on the side
A
B
AB
A
B
. Suppose that
B
B
B
,
K
K
K
,
L
L
L
,
C
C
C
are concyclic,
∠
A
K
D
=
∠
B
C
K
\angle AKD = \angle BCK
∠
A
KD
=
∠
BC
K
and
∠
A
L
D
=
∠
B
C
L
\angle ALD = \angle BCL
∠
A
L
D
=
∠
BC
L
. Prove that
A
K
=
A
L
AK = AL
A
K
=
A
L
.
3
1
Hide problems
Isometric conjugate
In triangle
A
B
C
,
A
B
>
A
C
,
I
ABC,AB>AC,I
A
BC
,
A
B
>
A
C
,
I
is the incenter,
A
M
AM
A
M
is the midline. The line crosses
I
I
I
and is perpendicular to
B
C
BC
BC
intersect
A
M
AM
A
M
at point
L
L
L
, and the symmetry of
I
I
I
with respect to point
A
A
A
is
J
J
J
Prove that:
∠
A
B
J
=
∠
L
B
I
\angle ABJ= \angle LBI
∠
A
B
J
=
∠
L
B
I
.
2
1
Hide problems
volleyball match
Let
n
n
n
be a positive integer. There are
3
n
3n
3
n
women's volleyball teams in the tournament, with no more than one match between every two teams (there are no ties in volleyball). We know that there are
3
n
2
3n^2
3
n
2
games played in this tournament. Proof: There exists a team with at least
n
4
\frac{n}{4}
4
n
win and
n
4
\frac{n}{4}
4
n
loss
1
1
Hide problems
a real sequence
Consider all the real sequences
x
0
,
x
1
,
⋯
,
x
100
x_0,x_1,\cdots,x_{100}
x
0
,
x
1
,
⋯
,
x
100
satisfying the following two requirements: (1)
x
0
=
0
x_0=0
x
0
=
0
; (2)For any integer
i
,
1
≤
i
≤
100
i,1\leq i\leq 100
i
,
1
≤
i
≤
100
,we have
1
≤
x
i
−
x
i
−
1
≤
2
1\leq x_i-x_{i-1}\leq 2
1
≤
x
i
−
x
i
−
1
≤
2
. Find the greatest positive integer
k
≤
100
k\leq 100
k
≤
100
,so that for any sequence
x
0
,
x
1
,
⋯
,
x
100
x_0,x_1,\cdots,x_{100}
x
0
,
x
1
,
⋯
,
x
100
like this,we have
x
k
+
x
k
+
1
+
⋯
+
x
100
≥
x
0
+
x
1
+
⋯
+
x
k
−
1
.
x_k+x_{k+1}+\cdots+x_{100}\geq x_0+x_1+\cdots+x_{k-1}.
x
k
+
x
k
+
1
+
⋯
+
x
100
≥
x
0
+
x
1
+
⋯
+
x
k
−
1
.
4
1
Hide problems
three consecutive positive integers
Given a prime number
p
≥
5
p\ge 5
p
≥
5
. Find the number of distinct remainders modulus
p
p
p
of the product of three consecutive positive integers.