MathDB
Problems
Contests
National and Regional Contests
China Contests
China Northern MO
2020 China Northern MO
2020 China Northern MO
Part of
China Northern MO
Subcontests
(10)
BP5
1
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Sets Whose Intersection Contains Relatively Prime Elements
It is known that subsets
A
1
,
A
2
,
⋯
,
A
n
A_1,A_2, \cdots , A_n
A
1
,
A
2
,
⋯
,
A
n
of set
I
=
{
1
,
2
,
⋯
,
101
}
I=\{1,2,\cdots ,101\}
I
=
{
1
,
2
,
⋯
,
101
}
satisfy the following condition
For any
i
,
j
(
1
≤
i
<
j
≤
n
)
, there exists
a
,
b
∈
A
i
∩
A
j
so that
(
a
,
b
)
=
1
\text{For any } i,j \text{ } (1 \leq i < j \leq n) \text{, there exists } a,b \in A_i \cap A_j \text{ so that } (a,b)=1
For any
i
,
j
(
1
≤
i
<
j
≤
n
)
, there exists
a
,
b
∈
A
i
∩
A
j
so that
(
a
,
b
)
=
1
Determine the maximum positive integer
n
n
n
.*
(
a
,
b
)
(a,b)
(
a
,
b
)
means
gcd
(
a
,
b
)
\gcd (a,b)
g
cd
(
a
,
b
)
BP4
1
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Cyclic with Incenter and Orthocenter
In
△
A
B
C
\triangle ABC
△
A
BC
,
∠
B
A
C
=
6
0
∘
\angle BAC = 60^{\circ}
∠
B
A
C
=
6
0
∘
, point
D
D
D
lies on side
B
C
BC
BC
,
O
1
O_1
O
1
and
O
2
O_2
O
2
are the centers of the circumcircles of
△
A
B
D
\triangle ABD
△
A
B
D
and
△
A
C
D
\triangle ACD
△
A
C
D
, respectively. Lines
B
O
1
BO_1
B
O
1
and
C
O
2
CO_2
C
O
2
intersect at point
P
P
P
. If
I
I
I
is the incenter of
△
A
B
C
\triangle ABC
△
A
BC
and
H
H
H
is the orthocenter of
△
P
B
C
\triangle PBC
△
PBC
, then prove that the four points
B
,
C
,
I
,
H
B,C,I,H
B
,
C
,
I
,
H
are on the same circle.
BP3
1
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19 divides 1+2^n+3^n+4^n
Are there infinitely many positive integers
n
n
n
such that
19
∣
1
+
2
n
+
3
n
+
4
n
19|1+2^n+3^n+4^n
19∣1
+
2
n
+
3
n
+
4
n
? Justify your claim.
BP2
1
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Non-symmetric Inequality to Prove
Given
a
,
b
,
c
∈
R
a,b,c \in \mathbb{R}
a
,
b
,
c
∈
R
satisfying
a
+
b
+
c
=
a
2
+
b
2
+
c
2
=
1
a+b+c=a^2+b^2+c^2=1
a
+
b
+
c
=
a
2
+
b
2
+
c
2
=
1
, show that
−
1
4
≤
a
b
≤
4
9
\frac{-1}{4} \leq ab \leq \frac{4}{9}
4
−
1
≤
ab
≤
9
4
.
BP1
1
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Inequality on Positive Reals
For all positive real numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
, prove that
a
3
+
b
3
a
2
−
a
b
+
b
2
+
b
3
+
c
3
b
2
−
b
c
+
c
2
+
c
3
+
a
3
c
2
−
c
a
+
a
2
≥
2
(
a
2
+
b
2
+
c
2
)
\frac{a^3+b^3}{ \sqrt{a^2-ab+b^2} } + \frac{b^3+c^3}{ \sqrt{b^2-bc+c^2} } + \frac{c^3+a^3}{ \sqrt{c^2-ca+a^2} } \geq 2(a^2+b^2+c^2)
a
2
−
ab
+
b
2
a
3
+
b
3
+
b
2
−
b
c
+
c
2
b
3
+
c
3
+
c
2
−
c
a
+
a
2
c
3
+
a
3
≥
2
(
a
2
+
b
2
+
c
2
)
P5
1
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(a+1)/2-digit number of 2s and 0s
Find all positive integers
a
a
a
so that for any
⌊
a
+
1
2
⌋
\left \lfloor \frac{a+1}{2} \right \rfloor
⌊
2
a
+
1
⌋
-digit number that is composed of only digits
0
0
0
and
2
2
2
(where
0
0
0
cannot be the first digit) is not a multiple of
a
a
a
.
P4
1
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Moving a Piece in a 20x20 Board
Two students
A
A
A
and
B
B
B
play a game on a
20
x
20
20 \text{ x } 20
20
x
20
chessboard. It is known that two squares are said to be adjacent if the two squares have a common side. At the beginning, there is a chess piece in a certain square of the chessboard. Given that
A
A
A
will be the first one to move the chess piece,
A
A
A
and
B
B
B
will alternately move this chess piece to an adjacent square. Also, the common side of any pair of adjacent squares can only be passed once. If the opponent cannot move anymore, then he will be declared the winner (to clarify since the wording wasn’t that good, you lose if you can’t move). Who among
A
A
A
and
B
B
B
has a winning strategy? Justify your claim.
P3
1
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Complete Residue System
A set of
k
k
k
integers is said to be a complete residue system modulo
k
k
k
if no two of its elements are congruent modulo
k
k
k
. Find all positive integers
m
m
m
so that there are infinitely many positive integers
n
n
n
wherein
{
1
n
,
2
n
,
…
,
m
n
}
\{ 1^n,2^n, \dots , m^n \}
{
1
n
,
2
n
,
…
,
m
n
}
is a complete residue system modulo
m
m
m
.
P2
1
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Weird angle is 30 degrees
In
△
A
B
C
\triangle ABC
△
A
BC
,
A
B
>
A
C
AB>AC
A
B
>
A
C
. Let
O
O
O
and
I
I
I
be the circumcenter and incenter respectively. Prove that if
∠
A
I
O
=
3
0
∘
\angle AIO = 30^{\circ}
∠
A
I
O
=
3
0
∘
, then
∠
A
B
C
=
6
0
∘
\angle ABC = 60^{\circ}
∠
A
BC
=
6
0
∘
.
P1
1
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Unbounded sum of sequence
The function
f
(
x
)
=
x
2
+
sin
x
f(x)=x^2+ \sin x
f
(
x
)
=
x
2
+
sin
x
and the sequence of positive numbers
{
a
n
}
\{ a_n \}
{
a
n
}
satisfy
a
1
=
1
a_1=1
a
1
=
1
,
f
(
a
n
)
=
a
n
−
1
f(a_n)=a_{n-1}
f
(
a
n
)
=
a
n
−
1
, where
n
≥
2
n \geq 2
n
≥
2
. Prove that there exists a positive integer
n
n
n
such that
a
1
+
a
2
+
⋯
+
a
n
>
2020
a_1+a_2+ \dots + a_n > 2020
a
1
+
a
2
+
⋯
+
a
n
>
2020
.