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Problems
Contests
National and Regional Contests
China Contests
China Girls Math Olympiad
2016 China Girls Math Olympiad
2016 China Girls Math Olympiad
Part of
China Girls Math Olympiad
Subcontests
(8)
7
1
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Prove equal fractions
In acute triangle
A
B
C
,
A
B
<
A
C
ABC, AB<AC
A
BC
,
A
B
<
A
C
,
I
I
I
is its incenter,
D
D
D
is the foot of perpendicular from
I
I
I
to
B
C
BC
BC
, altitude
A
H
AH
A
H
meets
B
I
,
C
I
BI,CI
B
I
,
C
I
at
P
,
Q
P,Q
P
,
Q
respectively. Let
O
O
O
be the circumcenter of
△
I
P
Q
\triangle IPQ
△
I
PQ
, extend
A
O
AO
A
O
to meet
B
C
BC
BC
at
L
L
L
. Circumcircle of
△
A
I
L
\triangle AIL
△
A
I
L
meets
B
C
BC
BC
again at
N
N
N
. Prove that
B
D
C
D
=
B
N
C
N
\frac{BD}{CD}=\frac{BN}{CN}
C
D
B
D
=
CN
BN
.
8
1
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Inequality on number of points on segment
Let
Q
\mathbb{Q}
Q
be the set of rational numbers,
Z
\mathbb{Z}
Z
be the set of integers. On the coordinate plane, given positive integer
m
m
m
, define
A
m
=
{
(
x
,
y
)
∣
x
,
y
∈
Q
,
x
y
≠
0
,
x
y
m
∈
Z
}
.
A_m = \left\{ (x,y)\mid x,y\in\mathbb{Q}, xy\neq 0, \frac{xy}{m}\in \mathbb{Z}\right\}.
A
m
=
{
(
x
,
y
)
∣
x
,
y
∈
Q
,
x
y
=
0
,
m
x
y
∈
Z
}
.
For segment
M
N
MN
MN
, define
f
m
(
M
N
)
f_m(MN)
f
m
(
MN
)
as the number of points on segment
M
N
MN
MN
belonging to set
A
m
A_m
A
m
.Find the smallest real number
λ
\lambda
λ
, such that for any line
l
l
l
on the coordinate plane, there exists a constant
β
(
l
)
\beta (l)
β
(
l
)
related to
l
l
l
, satisfying: for any two points
M
,
N
M,N
M
,
N
on
l
l
l
,
f
2016
(
M
N
)
≤
λ
f
2015
(
M
N
)
+
β
(
l
)
f_{2016}(MN)\le \lambda f_{2015}(MN)+\beta (l)
f
2016
(
MN
)
≤
λ
f
2015
(
MN
)
+
β
(
l
)
6
1
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Letters in grid
Find the greatest positive integer
m
m
m
, such that one of the
4
4
4
letters
C
,
G
,
M
,
O
C,G,M,O
C
,
G
,
M
,
O
can be placed in each cell of a table with
m
m
m
rows and
8
8
8
columns, and has the following property: For any two distinct rows in the table, there exists at most one column, such that the entries of these two rows in such a column are the same letter.
5
1
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Chinese Girls Mathematical Olympiad 2016, Problem 5
Define a sequence
{
a
n
}
\{a_n\}
{
a
n
}
by
S
1
=
1
,
S
n
+
1
=
(
2
+
S
n
)
2
4
+
S
n
(
n
=
1
,
2
,
3
,
⋯
)
.
S_1=1,\ S_{n+1}=\frac{(2+S_n)^2}{ 4+S_n} (n=1,\ 2,\ 3,\ \cdots).
S
1
=
1
,
S
n
+
1
=
4
+
S
n
(
2
+
S
n
)
2
(
n
=
1
,
2
,
3
,
⋯
)
.
Where
S
n
S_n
S
n
the sum of first
n
n
n
terms of sequence
{
a
n
}
\{a_n\}
{
a
n
}
. For any positive integer
n
n
n
,prove that
a
n
≥
4
9
n
+
7
.
a_{n}\ge \frac{4}{\sqrt{9n+7}}.
a
n
≥
9
n
+
7
4
.
2
1
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Sum of length of chords equal another
In
△
A
B
C
,
B
C
=
a
,
C
A
=
b
,
A
B
=
c
,
\triangle ABC, BC=a, CA=b, AB=c,
△
A
BC
,
BC
=
a
,
C
A
=
b
,
A
B
=
c
,
and
Γ
\Gamma
Γ
is its circumcircle.
(
1
)
(1)
(
1
)
Determine a necessary and sufficient condition on
a
,
b
a,b
a
,
b
and
c
c
c
if there exists a unique point
P
(
P
≠
B
,
P
≠
C
)
P(P\neq B, P\neq C)
P
(
P
=
B
,
P
=
C
)
on the arc
B
C
BC
BC
of
Γ
\Gamma
Γ
not passing through point
A
A
A
such that
P
A
=
P
B
+
P
C
PA=PB+PC
P
A
=
PB
+
PC
.
(
2
)
(2)
(
2
)
Let
P
P
P
be the unique point stated in
(
1
)
(1)
(
1
)
. If
A
P
AP
A
P
bisects
B
C
BC
BC
, prove that
∠
B
A
C
<
6
0
∘
\angle BAC<60^{\circ}
∠
B
A
C
<
6
0
∘
.
1
1
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Switching two cards in two boxes
Let
n
≥
3
n\ge 3
n
≥
3
be an integer. Put
n
2
n^2
n
2
cards, each labelled
1
,
2
,
…
,
n
2
1,2,\ldots ,n^2
1
,
2
,
…
,
n
2
respectively, in any order into
n
n
n
empty boxes such that there are exactly
n
n
n
cards in each box. One can perform the following operation: one first selects
2
2
2
boxes, takes out any
2
2
2
cards from each of the selected boxes, and then return the cards to the other selected box. Prove that, for any initial order of the
n
2
n^2
n
2
cards in the boxes, one can perform the operation finitely many times such that the labelled numbers in each box are consecutive integers.
4
1
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Chinese Girls Mathematical Olympiad 2016, Problem 4
Let
n
n
n
is a positive integers ,
a
1
,
a
2
,
⋯
,
a
n
∈
{
0
,
1
,
⋯
,
n
}
a_1,a_2,\cdots,a_n\in\{0,1,\cdots,n\}
a
1
,
a
2
,
⋯
,
a
n
∈
{
0
,
1
,
⋯
,
n
}
. For the integer
j
j
j
(
1
≤
j
≤
n
)
(1\le j\le n)
(
1
≤
j
≤
n
)
,define
b
j
b_j
b
j
is the number of elements in the set
{
i
∣
i
∈
{
1
,
⋯
,
n
}
,
a
i
≥
j
}
\{i|i\in\{1,\cdots,n\},a_i\ge j\}
{
i
∣
i
∈
{
1
,
⋯
,
n
}
,
a
i
≥
j
}
.For example ：When
n
=
3
n=3
n
=
3
,if
a
1
=
1
,
a
2
=
2
,
a
3
=
1
a_1=1,a_2=2,a_3=1
a
1
=
1
,
a
2
=
2
,
a
3
=
1
,then
b
1
=
3
,
b
2
=
1
,
b
3
=
0
b_1=3,b_2=1,b_3=0
b
1
=
3
,
b
2
=
1
,
b
3
=
0
.
(
1
)
(1)
(
1
)
Prove that
∑
i
=
1
n
(
i
+
a
i
)
2
≥
∑
i
=
1
n
(
i
+
b
i
)
2
.
\sum_{i=1}^{n}(i+a_i)^2\ge \sum_{i=1}^{n}(i+b_i)^2.
i
=
1
∑
n
(
i
+
a
i
)
2
≥
i
=
1
∑
n
(
i
+
b
i
)
2
.
(
2
)
(2)
(
2
)
Prove that
∑
i
=
1
n
(
i
+
a
i
)
k
≥
∑
i
=
1
n
(
i
+
b
i
)
k
,
\sum_{i=1}^{n}(i+a_i)^k\ge \sum_{i=1}^{n}(i+b_i)^k,
i
=
1
∑
n
(
i
+
a
i
)
k
≥
i
=
1
∑
n
(
i
+
b
i
)
k
,
for the integer
k
≥
3.
k\ge 3.
k
≥
3.
3
1
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Chinese Girls Mathematical Olympiad 2016, Problem 3
Let
m
m
m
and
n
n
n
are relatively prime integers and
m
>
1
,
n
>
1
m>1,n>1
m
>
1
,
n
>
1
. Show that:There are positive integers
a
,
b
,
c
a,b,c
a
,
b
,
c
such that
m
a
=
1
+
n
b
c
m^a=1+n^bc
m
a
=
1
+
n
b
c
, and
n
n
n
and
c
c
c
are relatively prime.