MathDB
Problems
Contests
National and Regional Contests
China Contests
(China) National High School Mathematics League
2022 China Second Round A1
2022 China Second Round A1
Part of
(China) National High School Mathematics League
Subcontests
(4)
4
1
Hide problems
Filling the blank
Given
r
∈
R
r\in\mathbb{R}
r
∈
R
. Alice and Bob plays the following game: An equation with blank is written on the blackboard as below:
S
=
∣
□
−
□
∣
+
∣
□
−
□
∣
+
∣
□
−
□
∣
S=|\Box-\Box|+|\Box-\Box|+|\Box-\Box|
S
=
∣
□
−
□
∣
+
∣
□
−
□
∣
+
∣
□
−
□
∣
Every round, Alice choose a real number from
[
0
,
1
]
[0,1]
[
0
,
1
]
(not necessary to be different from the numbers chosen before) and Bob fill it in an empty box. After 6 rounds, every blank is filled and
S
S
S
is determined at the same time. If
S
≥
r
S\ge r
S
≥
r
then Alice wins, otherwise Bob wins. Find all
r
r
r
such that Alice can guarantee her victory.
2
1
Hide problems
Passing through midpoint
In acute triangle
△
A
B
C
\triangle ABC
△
A
BC
,
H
H
H
is the orthocenter,
B
D
BD
B
D
,
C
E
CE
CE
are altitudes.
M
M
M
is the midpoint of
B
C
BC
BC
.
P
P
P
,
Q
Q
Q
are on segment
B
M
BM
BM
,
D
E
DE
D
E
, respectively.
R
R
R
is on segment
P
Q
PQ
PQ
such that
B
P
E
Q
=
C
P
D
Q
=
P
R
Q
R
\frac{BP}{EQ}=\frac{CP}{DQ}=\frac{PR}{QR}
EQ
BP
=
D
Q
CP
=
QR
PR
. Suppose
L
L
L
is the orthocenter of
△
A
H
R
\triangle AHR
△
A
H
R
, then prove:
Q
M
QM
QM
passes through the midpoint of
R
L
RL
R
L
.
3
1
Hide problems
the existence of a set...or not?
Does there exist an infinite set
S
S
S
consisted of positive integers,so that for any
x
,
y
,
z
,
w
∈
S
,
x
<
y
,
z
<
w
x,y,z,w\in S,x<y,z<w
x
,
y
,
z
,
w
∈
S
,
x
<
y
,
z
<
w
,if
(
x
,
y
)
≠
(
z
,
w
)
(x,y)\ne (z,w)
(
x
,
y
)
=
(
z
,
w
)
,then
gcd
(
x
y
+
2022
,
z
w
+
2022
)
=
1
\gcd(xy+2022,zw+2022)=1
g
cd
(
x
y
+
2022
,
z
w
+
2022
)
=
1
?
1
1
Hide problems
find the maximum and minimum
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
are real numbers so that
a
≥
b
,
c
≥
d
a\geq b,c\geq d
a
≥
b
,
c
≥
d
,
∣
a
∣
+
2
∣
b
∣
+
3
∣
c
∣
+
4
∣
d
∣
=
1.
|a|+2|b|+3|c|+4|d|=1.
∣
a
∣
+
2∣
b
∣
+
3∣
c
∣
+
4∣
d
∣
=
1.
Let
P
=
(
a
−
b
)
(
b
−
c
)
(
c
−
d
)
P=(a-b)(b-c)(c-d)
P
=
(
a
−
b
)
(
b
−
c
)
(
c
−
d
)
,find the maximum and minimum value of
P
P
P
.