MathDB
Problems
Contests
National and Regional Contests
China Contests
(China) National High School Mathematics League
2023 China Second Round
2023 China Second Round
Part of
(China) National High School Mathematics League
Subcontests
(8)
7
1
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throw a Dice three times
We throw a dice three times and the numbers are
x
,
y
,
z
x,y,z
x
,
y
,
z
find out the possibility of
(
7
x
)
<
(
7
y
)
<
(
7
z
)
\binom{7}{x}<\binom{7}{y}<\binom{7}{z}
(
x
7
)
<
(
y
7
)
<
(
z
7
)
3
1
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Problem 3
Find the smallest positive integer
k
{k}
k
with the following properties
:
:{}{}{}{}{}
:
If each positive integer is arbitrarily colored red or blue
,
{}{}{},
,
there may be
9
{}{}{}{}9
9
distinct red positive integers
x
1
,
x
2
,
⋯
,
x
9
,
x_1,x_2,\cdots ,x_9,
x
1
,
x
2
,
⋯
,
x
9
,
satisfying
x
1
+
x
2
+
⋯
+
x
8
<
x
9
,
x_1+x_2+\cdots +x_8<x_9,
x
1
+
x
2
+
⋯
+
x
8
<
x
9
,
or there are
10
10{}{}{}{}{}{}
10
distinct blue positive integers
y
1
,
y
2
,
⋯
,
y
10
y_1,y_2,\cdots ,y_{10}
y
1
,
y
2
,
⋯
,
y
10
satisfiying
y
1
+
y
2
+
⋯
+
y
9
<
y
10
.
{y_1+y_2+\cdots +y_9<y_{10}}.
y
1
+
y
2
+
⋯
+
y
9
<
y
10
.
5
1
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China Second Round Olympiad 2023 Test 1 Q 5
Find the sum of the smallest 20 positive real solutions of the equation
sin
x
=
cos
2
x
.
\sin x=\cos 2x .
sin
x
=
cos
2
x
.
6
1
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China Second Round Olympiad 2023 Test 1 Q 6
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be the lengths of the three sides of a triangle and
a
,
b
a,b
a
,
b
be the two roots of the equation
a
x
2
−
b
x
+
c
=
0
ax^2-bx+c=0
a
x
2
−
b
x
+
c
=
0
(
a
<
b
)
.
(a<b) .
(
a
<
b
)
.
Find the value range of
a
+
b
−
c
.
a+b-c .
a
+
b
−
c
.
11
1
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China Second Round Olympiad 2023 Test 1 Q 11
Find all real numbers
t
t
t
not less than
1
1
1
that satisfy the following requirements: for any
a
,
b
∈
[
−
1
,
t
]
a,b\in [-1,t]
a
,
b
∈
[
−
1
,
t
]
, there always exists
c
,
d
∈
[
−
1
,
t
]
c,d \in [-1,t ]
c
,
d
∈
[
−
1
,
t
]
such that
(
a
+
c
)
(
b
+
d
)
=
1.
(a+c)(b+d)=1.
(
a
+
c
)
(
b
+
d
)
=
1.
1
2
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A problem of Complex numbers
We define a complex number
z
=
9
+
10
i
z=9+10i
z
=
9
+
10
i
please find the maximum of a positive integer
n
n
n
which satisfies
∣
z
n
∣
≤
2023
|z^n|\leq2023
∣
z
n
∣
≤
2023
Fixed Circumcenter
Let
A
,
B
A,B
A
,
B
be two fixed points on a plane and
Ω
\Omega
Ω
a fixed semicircle arc with diameter
A
B
AB
A
B
. Let
T
T
T
be another fixed point on
Ω
\Omega
Ω
, and
ω
\omega
ω
a fixed circle that passes through
A
A
A
and
T
T
T
and has its center in
Δ
A
B
T
\Delta ABT
Δ
A
BT
. Let
P
P
P
be a moving point on the arc
T
B
TB
TB
(endpoints excluded), and
C
,
D
C,D
C
,
D
be two moving points on
ω
\omega
ω
such that
C
C
C
lies on segment
A
P
AP
A
P
,
C
,
D
C,D
C
,
D
lies on different sides of line
A
B
AB
A
B
and
C
D
⊥
A
B
CD\ \bot \ AB
C
D
⊥
A
B
. Denote the circumcenter of
Δ
C
D
P
\Delta CDP
Δ
C
D
P
of
K
K
K
. Prove that (i)
K
K
K
lies on the circumcircle of
Δ
T
D
P
\Delta TDP
Δ
T
D
P
. (ii)
K
K
K
is a fixed point.
4
2
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A problem of Vectors
if three non-zero vectors on a plane
a
⃗
,
b
⃗
,
c
⃗
\vec{a},\vec{b},\vec{c}
a
,
b
,
c
satisfy: (a)
a
⃗
⊥
b
⃗
\vec{a}\bot\vec{b}
a
⊥
b
(b)
b
⃗
⋅
c
⃗
=
2
∣
a
⃗
∣
\vec{b}\cdot\vec{c}=2|\vec{a}|
b
⋅
c
=
2∣
a
∣
(c)
c
⃗
⋅
a
⃗
=
3
∣
b
⃗
∣
\vec{c}\cdot\vec{a}=3|\vec{b}|
c
⋅
a
=
3∣
b
∣
find out the minimum of
∣
c
⃗
∣
|\vec{c}|
∣
c
∣
On the Ratio of Products of Sum of Each Row and Each Column
Let
a
=
1
+
1
0
−
4
a=1+10^{-4}
a
=
1
+
1
0
−
4
. Consider some
2023
×
2023
2023\times 2023
2023
×
2023
matrix with each entry a real in
[
1
,
a
]
[1,a]
[
1
,
a
]
. Let
x
i
x_i
x
i
be the sum of the elements of the
i
i
i
-th row and
y
i
y_i
y
i
be the sum of the elements of the
i
i
i
-th column for each integer
i
∈
[
1
,
n
]
i\in [1,n]
i
∈
[
1
,
n
]
. Find the maximum possible value of
y
1
y
2
⋯
y
2023
x
1
x
2
⋯
x
2023
\dfrac{y_1y_2\cdots y_{2023}}{x_1x_2\cdots x_{2023}}
x
1
x
2
⋯
x
2023
y
1
y
2
⋯
y
2023
(the answer may be expressed in terms of
a
a
a
).
2
2
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exponents and logarithms
if a,b∈R+,
a
log
b
=
2
a^{\log b}=2
a
l
o
g
b
=
2
,
a
log
a
b
log
b
=
5
a^{\log a}b^{\log b}=5
a
l
o
g
a
b
l
o
g
b
=
5
,find out
(
a
b
)
log
a
b
(ab)^{\log ab}
(
ab
)
l
o
g
ab
$n^2/2^n \neq m^2/2^m$ for Every $m$
For some positive integer
n
n
n
,
n
n
n
is considered a
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
u
n
i
q
u
e
<
/
s
p
a
n
>
<span class='latex-bold'>unique</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
u
ni
q
u
e
<
/
s
p
an
>
number if for any other positive integer
m
≠
n
m\neq n
m
=
n
,
{
2
n
n
2
}
≠
{
2
m
m
2
}
\{\dfrac{2^n}{n^2}\}\neq\{\dfrac{2^m}{m^2}\}
{
n
2
2
n
}
=
{
m
2
2
m
}
holds. Prove that there is an infinite list consisting of composite unique numbers whose elements are pairwise coprime.