MathDB
Problems
Contests
National and Regional Contests
China Contests
(China) National High School Mathematics League
2022 China Second Round A2
2022 China Second Round A2
Part of
(China) National High School Mathematics League
Subcontests
(4)
4
1
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NT about sequence again
k
>
2
k>2
k
>
2
is an integer.
a
0
,
a
1
,
.
.
.
a_0,a_1,...
a
0
,
a
1
,
...
is an integer sequence such that
a
0
=
0
a_0=0
a
0
=
0
,
a
n
+
1
=
k
a
n
−
a
n
−
1
a_{n+1}=ka_n-a_{n-1}
a
n
+
1
=
k
a
n
−
a
n
−
1
. Prove that for any positive integer
m
m
m
,
(
2
m
)
!
∣
a
1
a
2
.
.
.
a
3
m
(2m)!|a_1a_2...a_{3m}
(
2
m
)!
∣
a
1
a
2
...
a
3
m
.
3
1
Hide problems
K_500s cover complete graph
S
=
{
1
,
2
,
.
.
.
,
N
}
S=\{1,2,...,N\}
S
=
{
1
,
2
,
...
,
N
}
.
A
1
,
A
2
,
A
3
,
A
4
⊆
S
A_1,A_2,A_3,A_4\subseteq S
A
1
,
A
2
,
A
3
,
A
4
⊆
S
, each having cardinality
500
500
500
.
∀
x
,
y
∈
S
\forall x,y\in S
∀
x
,
y
∈
S
,
∃
i
∈
{
1
,
2
,
3
,
4
}
\exists i\in\{1,2,3,4\}
∃
i
∈
{
1
,
2
,
3
,
4
}
,
x
,
y
∈
A
i
x,y\in A_i
x
,
y
∈
A
i
. Determine the maximal value of
N
N
N
.
2
1
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Two circles and many points
A
,
B
,
C
,
D
,
E
A,B,C,D,E
A
,
B
,
C
,
D
,
E
are points on a circle
ω
\omega
ω
, satisfying
A
B
=
B
D
AB=BD
A
B
=
B
D
,
B
C
=
C
E
BC=CE
BC
=
CE
.
A
C
AC
A
C
meets
B
E
BE
BE
at
P
P
P
.
Q
Q
Q
is on
D
E
DE
D
E
such that
B
E
/
/
A
Q
BE//AQ
BE
//
A
Q
. Suppose
⊙
(
A
P
Q
)
\odot(APQ)
⊙
(
A
PQ
)
intersects
ω
\omega
ω
again at
T
T
T
.
A
′
A'
A
′
is the reflection of
A
A
A
wrt
B
C
BC
BC
. Prove that
A
′
B
P
T
A'BPT
A
′
BPT
lies on the same circle.
1
1
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Inequality problem with max and min
a
1
,
a
2
,
.
.
.
,
a
9
a_1,a_2,...,a_9
a
1
,
a
2
,
...
,
a
9
are nonnegative reals with sum
1
1
1
. Define
S
S
S
and
T
T
T
as below:
S
=
min
{
a
1
,
a
2
}
+
2
min
{
a
2
,
a
3
}
+
.
.
.
+
9
min
{
a
9
,
a
1
}
S=\min\{a_1,a_2\}+2\min\{a_2,a_3\}+...+9\min\{a_9,a_1\}
S
=
min
{
a
1
,
a
2
}
+
2
min
{
a
2
,
a
3
}
+
...
+
9
min
{
a
9
,
a
1
}
T
=
max
{
a
1
,
a
2
}
+
2
max
{
a
2
,
a
3
}
+
.
.
.
+
9
max
{
a
9
,
a
1
}
T=\max\{a_1,a_2\}+2\max\{a_2,a_3\}+...+9\max\{a_9,a_1\}
T
=
max
{
a
1
,
a
2
}
+
2
max
{
a
2
,
a
3
}
+
...
+
9
max
{
a
9
,
a
1
}
When
S
S
S
reaches its maximum, find all possible values of
T
T
T
.