MathDB
Problems
Contests
National and Regional Contests
China Contests
China Girls Math Olympiad
2020 China Girls Math Olympiad
2020 China Girls Math Olympiad
Part of
China Girls Math Olympiad
Subcontests
(8)
8
1
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Determine the number of (a_1,a_2,\cdots,a_m)
Let
n
n
n
be a given positive integer. Let
N
+
\mathbb{N}_+
N
+
denote the set of all positive integers.Determine the number of all finite lists
(
a
1
,
a
2
,
⋯
,
a
m
)
(a_1,a_2,\cdots,a_m)
(
a
1
,
a
2
,
⋯
,
a
m
)
such that: (1)
m
∈
N
+
m\in \mathbb{N}_+
m
∈
N
+
and
a
1
,
a
2
,
⋯
,
a
m
∈
N
+
a_1,a_2,\cdots,a_m\in \mathbb{N}_+
a
1
,
a
2
,
⋯
,
a
m
∈
N
+
and
a
1
+
a
2
+
⋯
+
a
m
=
n
a_1+a_2+\cdots+a_m=n
a
1
+
a
2
+
⋯
+
a
m
=
n
. (2) The number of all pairs of integers
(
i
,
j
)
(i,j)
(
i
,
j
)
satisfying
1
≤
i
<
j
≤
m
1\leq i<j\leq m
1
≤
i
<
j
≤
m
and
a
i
>
a
j
a_i>a_j
a
i
>
a
j
is even.For example, when
n
=
4
n=4
n
=
4
, the number of all such lists
(
a
1
,
a
2
,
⋯
,
a
m
)
(a_1,a_2,\cdots,a_m)
(
a
1
,
a
2
,
⋯
,
a
m
)
is
6
6
6
, and these lists are
(
4
)
,
(4),
(
4
)
,
(
1
,
3
)
,
(1,3),
(
1
,
3
)
,
(
2
,
2
)
,
(2,2),
(
2
,
2
)
,
(
1
,
1
,
2
)
,
(1,1,2),
(
1
,
1
,
2
)
,
(
2
,
1
,
1
)
,
(2,1,1),
(
2
,
1
,
1
)
,
(
1
,
1
,
1
,
1
)
(1,1,1,1)
(
1
,
1
,
1
,
1
)
.
6
1
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a number theory problem
Let
p
,
q
p, q
p
,
q
be integers and
p
,
q
>
1
p, q > 1
p
,
q
>
1
,
g
c
d
(
p
,
6
q
)
=
1
gcd(p, \,6q)=1
g
c
d
(
p
,
6
q
)
=
1
. Prove that:
∑
k
=
1
q
−
1
⌊
p
k
q
⌋
2
≡
2
p
∑
k
=
1
q
−
1
k
⌊
p
k
q
⌋
(
m
o
d
q
−
1
)
\sum_{k=1}^{q-1}\left \lfloor \frac{pk}{q}\right\rfloor^2 \equiv 2p \sum_{k=1}^{q-1}k\left\lfloor \frac{pk}{q} \right\rfloor (mod \, q-1)
k
=
1
∑
q
−
1
⌊
q
p
k
⌋
2
≡
2
p
k
=
1
∑
q
−
1
k
⌊
q
p
k
⌋
(
m
o
d
q
−
1
)
5
1
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Find all the real number sequences
Find all the real number sequences
{
b
n
}
n
≥
1
\{b_n\}_{n \geq 1}
{
b
n
}
n
≥
1
and
{
c
n
}
n
≥
1
\{c_n\}_{n \geq 1}
{
c
n
}
n
≥
1
that satisfy the following conditions: (i) For any positive integer
n
n
n
,
b
n
≤
c
n
b_n \leq c_n
b
n
≤
c
n
; (ii) For any positive integer
n
n
n
,
b
n
+
1
b_{n+1}
b
n
+
1
and
c
n
+
1
c_{n+1}
c
n
+
1
is the two roots of the equation
x
2
+
b
n
x
+
c
n
=
0
x^2+b_nx+c_n=0
x
2
+
b
n
x
+
c
n
=
0
.
7
1
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Perpendicular to line through orthocenter and incenter
Let
O
O
O
be the circumcenter of triangle
△
A
B
C
\triangle ABC
△
A
BC
, where
∠
B
A
C
=
12
0
∘
\angle BAC=120^{\circ}
∠
B
A
C
=
12
0
∘
. The tangent at
A
A
A
to
(
A
B
C
)
(ABC)
(
A
BC
)
meets the tangents at
B
,
C
B,C
B
,
C
at
(
A
B
C
)
(ABC)
(
A
BC
)
at points
P
,
Q
P,Q
P
,
Q
respectively. Let
H
,
I
H,I
H
,
I
be the orthocenter and incenter of
△
O
P
Q
\triangle OPQ
△
OPQ
respectively. Define
M
,
N
M,N
M
,
N
as the midpoints of arc \overarc{BAC} and
O
I
OI
O
I
respectively, and let
M
N
MN
MN
meet
(
A
B
C
)
(ABC)
(
A
BC
)
again at
D
D
D
. Prove that
A
D
AD
A
D
is perpendicular to
H
I
HI
H
I
.
3
1
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At least 10 tall guys no matter the partition
There are
3
3
3
classes with
n
n
n
students in each class, and the heights of all
3
n
3n
3
n
students are pairwise distinct. Partition the students into groups of
3
3
3
such that in each group, there is one student from each class. In each group, call the tallest student the tall guy. Suppose that for any partition of the students, there are at least 10 tall guys in each class, prove that the minimum value of
n
n
n
is
40
40
40
.
4
1
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Gcd of factorial minus 1
Let
p
,
q
p,q
p
,
q
be primes, where
p
>
q
p>q
p
>
q
. Define
t
=
gcd
(
p
!
−
1
,
q
!
−
1
)
t=\gcd(p!-1,q!-1)
t
=
g
cd
(
p
!
−
1
,
q
!
−
1
)
. Prove that
t
≤
p
p
3
t\le p^{\frac{p}{3}}
t
≤
p
3
p
.
1
1
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a geometry problem
In the quadrilateral
A
B
C
D
ABCD
A
BC
D
,
A
B
=
A
D
AB=AD
A
B
=
A
D
,
C
B
=
C
D
CB=CD
CB
=
C
D
,
∠
A
B
C
=
9
0
∘
\angle ABC =90^\circ
∠
A
BC
=
9
0
∘
.
E
E
E
,
F
F
F
are on
A
B
AB
A
B
,
A
D
AD
A
D
and
P
P
P
,
Q
Q
Q
are on
E
F
EF
EF
(
P
P
P
is between
E
,
Q
E, Q
E
,
Q
), satisfy
A
E
E
P
=
A
F
F
Q
\frac{AE}{EP}=\frac{AF}{FQ}
EP
A
E
=
FQ
A
F
.
X
,
Y
X, Y
X
,
Y
are on
C
P
,
C
Q
CP, CQ
CP
,
CQ
that satisfy
B
X
⊥
C
P
,
D
Y
⊥
C
Q
BX \perp CP, DY \perp CQ
BX
⊥
CP
,
D
Y
⊥
CQ
. Prove that
X
,
P
,
Q
,
Y
X, P, Q, Y
X
,
P
,
Q
,
Y
are concyclic.
2
1
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Find Maximum Value
Let
n
n
n
be an integer and
n
≥
2
n \geq 2
n
≥
2
,
x
1
,
x
2
,
⋯
,
x
n
x_1, x_2, \cdots , x_n
x
1
,
x
2
,
⋯
,
x
n
are arbitrary real number, find the maximum value of
2
∑
1
≤
i
<
j
≤
n
⌊
x
i
x
j
⌋
−
(
n
−
1
)
∑
i
=
1
n
⌊
x
i
2
⌋
2\sum_{1\leq i<j \leq n}\left \lfloor x_ix_j \right \rfloor-\left ( n-1 \right )\sum_{i=1}^{n}\left \lfloor x_i^2 \right \rfloor
2
1
≤
i
<
j
≤
n
∑
⌊
x
i
x
j
⌋
−
(
n
−
1
)
i
=
1
∑
n
⌊
x
i
2
⌋