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Problems
Contests
National and Regional Contests
China Contests
China National Olympiad
2020 China National Olympiad
2020 China National Olympiad
Part of
China National Olympiad
Subcontests
(6)
6
1
Hide problems
Swapping coefficients gives roots
Does there exist positive reals
a
0
,
a
1
,
…
,
a
19
a_0, a_1,\ldots ,a_{19}
a
0
,
a
1
,
…
,
a
19
, such that the polynomial
P
(
x
)
=
x
20
+
a
19
x
19
+
…
+
a
1
x
+
a
0
P(x)=x^{20}+a_{19}x^{19}+\ldots +a_1x+a_0
P
(
x
)
=
x
20
+
a
19
x
19
+
…
+
a
1
x
+
a
0
does not have any real roots, yet all polynomials formed from swapping any two coefficients
a
i
,
a
j
a_i,a_j
a
i
,
a
j
has at least one real root?
4
1
Hide problems
Common intersection of arcs
Find the largest positive constant
C
C
C
such that the following is satisfied: Given
n
n
n
arcs (containing their endpoints)
A
1
,
A
2
,
…
,
A
n
A_1,A_2,\ldots ,A_n
A
1
,
A
2
,
…
,
A
n
on the circumference of a circle, where among all sets of three arcs
(
A
i
,
A
j
,
A
k
)
(A_i,A_j,A_k)
(
A
i
,
A
j
,
A
k
)
(
1
≤
i
<
j
<
k
≤
n
)
(1\le i< j< k\le n)
(
1
≤
i
<
j
<
k
≤
n
)
, at least half of them has
A
i
∩
A
j
∩
A
k
A_i\cap A_j\cap A_k
A
i
∩
A
j
∩
A
k
nonempty, then there exists
l
>
C
n
l>Cn
l
>
C
n
, such that we can choose
l
l
l
arcs among
A
1
,
A
2
,
…
,
A
n
A_1,A_2,\ldots ,A_n
A
1
,
A
2
,
…
,
A
n
, whose intersection is nonempty.
5
1
Hide problems
Perfect square in sequence
Given any positive integer
c
c
c
, denote
p
(
c
)
p(c)
p
(
c
)
as the largest prime factor of
c
c
c
. A sequence
{
a
n
}
\{a_n\}
{
a
n
}
of positive integers satisfies
a
1
>
1
a_1>1
a
1
>
1
and
a
n
+
1
=
a
n
+
p
(
a
n
)
a_{n+1}=a_n+p(a_n)
a
n
+
1
=
a
n
+
p
(
a
n
)
for all
n
≥
1
n\ge 1
n
≥
1
. Prove that there must exist at least one perfect square in sequence
{
a
n
}
\{a_n\}
{
a
n
}
.
2
1
Hide problems
China Mathematical Olympiad 2020 Q2
In triangle
A
B
C
ABC
A
BC
,
A
B
>
A
C
.
AB>AC.
A
B
>
A
C
.
The bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
meets
B
C
BC
BC
at
D
.
D.
D
.
P
P
P
is on line
D
A
,
DA,
D
A
,
such that
A
A
A
lies between
P
P
P
and
D
D
D
.
P
Q
PQ
PQ
is tangent to
⊙
(
A
B
D
)
\odot(ABD)
⊙
(
A
B
D
)
at
Q
.
Q.
Q
.
P
R
PR
PR
is tangent to
⊙
(
A
C
D
)
\odot(ACD)
⊙
(
A
C
D
)
at
R
.
R.
R
.
C
Q
CQ
CQ
meets
B
R
BR
BR
at
K
.
K.
K
.
The line parallel to
B
C
BC
BC
and passing through
K
K
K
meets
Q
D
,
A
D
,
R
D
QD,AD,RD
Q
D
,
A
D
,
R
D
at
E
,
L
,
F
,
E,L,F,
E
,
L
,
F
,
respectively. Prove that
E
L
=
K
F
.
EL=KF.
E
L
=
K
F
.
3
1
Hide problems
2020 China Mathematics Olympiad
Let
S
S
S
be a set,
∣
S
∣
=
35
|S|=35
∣
S
∣
=
35
. A set
F
F
F
of mappings from
S
S
S
to itself is called to be satisfying property
P
(
k
)
P(k)
P
(
k
)
, if for any
x
,
y
∈
S
x,y\in S
x
,
y
∈
S
, there exist
f
1
,
⋯
,
f
k
∈
F
f_1, \cdots, f_k \in F
f
1
,
⋯
,
f
k
∈
F
(not necessarily different), such that
f
k
(
f
k
−
1
(
⋯
(
f
1
(
x
)
)
)
)
=
f
k
(
f
k
−
1
(
⋯
(
f
1
(
y
)
)
)
)
f_k(f_{k-1}(\cdots (f_1(x))))=f_k(f_{k-1}(\cdots (f_1(y))))
f
k
(
f
k
−
1
(
⋯
(
f
1
(
x
))))
=
f
k
(
f
k
−
1
(
⋯
(
f
1
(
y
))))
. Find the least positive integer
m
m
m
, such that if
F
F
F
satisfies property
P
(
2019
)
P(2019)
P
(
2019
)
, then it also satisfies property
P
(
m
)
P(m)
P
(
m
)
.
1
1
Hide problems
China Mathematical Olympiad 2020 Q1
Let
a
1
,
a
2
,
⋯
,
a
41
∈
R
,
a_1,a_2,\cdots,a_{41}\in\mathbb{R},
a
1
,
a
2
,
⋯
,
a
41
∈
R
,
such that
a
41
=
a
1
,
∑
i
=
1
40
a
i
=
0
,
a_{41}=a_1, \sum_{i=1}^{40}a_i=0,
a
41
=
a
1
,
∑
i
=
1
40
a
i
=
0
,
and for any
i
=
1
,
2
,
⋯
,
40
,
∣
a
i
−
a
i
+
1
∣
≤
1.
i=1,2,\cdots,40, |a_i-a_{i+1}|\leq 1.
i
=
1
,
2
,
⋯
,
40
,
∣
a
i
−
a
i
+
1
∣
≤
1.
Determine the greatest possible value of
(
1
)
a
10
+
a
20
+
a
30
+
a
40
;
(1)a_{10}+a_{20}+a_{30}+a_{40};
(
1
)
a
10
+
a
20
+
a
30
+
a
40
;
(
2
)
a
10
⋅
a
20
+
a
30
⋅
a
40
.
(2)a_{10}\cdot a_{20}+a_{30}\cdot a_{40}.
(
2
)
a
10
⋅
a
20
+
a
30
⋅
a
40
.