MathDB
Problems
Contests
National and Regional Contests
China Contests
China Northern MO
2013 China Northern MO
2013 China Northern MO
Part of
China Northern MO
Subcontests
(8)
8
1
Hide problems
3n people in a gathering
3
n
3n
3
n
(
n
≥
2
,
n
∈
N
n \ge 2, n \in N
n
≥
2
,
n
∈
N
) people attend a gathering, in which any two acquaintances have exactly
n
n
n
common acquaintances, and any two unknown people have exactly
2
n
2n
2
n
common acquaintances. If three people know each other, it is called a Taoyuan Group. (1) Find the number of all Taoyuan groups; (2) Prove that these
3
n
3n
3
n
people can be divided into three groups, with
n
n
n
people in each group, and the three people obtained by randomly selecting one person from each group constitute a Taoyuan group.Note: Acquaintance means that two people know each other, otherwise they are not acquaintances. Two people who know each other are called acquaintances.
4
1
Hide problems
13^k=a^2 +b^2
For positive integers
n
,
a
,
b
n,a,b
n
,
a
,
b
, if
n
=
a
2
+
b
2
n=a^2 +b^2
n
=
a
2
+
b
2
, and
a
a
a
and
b
b
b
are coprime, then the number pair
(
a
,
b
)
(a,b)
(
a
,
b
)
is called a square split of
n
n
n
(the order of
a
,
b
a, b
a
,
b
does not count). Prove that for any positive
k
k
k
, there are only two square splits of the integer
1
3
k
13^k
1
3
k
.
1
1
Hide problems
max n ,tangent of each interior angle in convex n-gon is an integer.
Find the largest positive integer
n
n
n
(
n
≥
3
n \ge 3
n
≥
3
), so that there is a convex
n
n
n
-gon, the tangent of each interior angle is an integer.
6
1
Hide problems
PO _|_BC wanted, circle passing thorugh A,C and tangent to median AM
As shown in figure , it is known that
M
M
M
is the midpoint of side
B
C
BC
BC
of
△
A
B
C
\vartriangle ABC
△
A
BC
.
⊙
O
\odot O
⊙
O
passes through points
A
,
C
A, C
A
,
C
and is tangent to
A
M
AM
A
M
. The extension of the segment
B
A
BA
B
A
intersects
⊙
O
\odot O
⊙
O
at point
D
D
D
. The lines
C
D
CD
C
D
and
M
A
MA
M
A
intersect at the point
P
P
P
. Prove that
P
O
⊥
B
C
PO \perp BC
PO
⊥
BC
. https://cdn.artofproblemsolving.com/attachments/8/a/da3570ec7eb0833c7a396e22ffac2bd8902186.png
3
1
Hide problems
fixed length wanted, circle passing through 2 fixed points
As shown in figure ,
A
,
B
A,B
A
,
B
are two fixed points of circle
⊙
O
\odot O
⊙
O
,
C
C
C
is the midpoint of the major arc
A
B
AB
A
B
,
D
D
D
is any point of the minor arc
A
B
AB
A
B
. Tangent at
D
D
D
intersects tangents at
A
,
B
A,B
A
,
B
at points
E
,
F
E,F
E
,
F
respectively. Segments
C
E
CE
CE
and
C
F
CF
CF
intersect chord
A
B
AB
A
B
at points
G
G
G
and
H
H
H
respectively. Prove that the length of line segment
G
H
GH
G
H
has a fixed value. https://cdn.artofproblemsolving.com/attachments/9/2/85227f169193f61e313293e9128f6ece2ff1f7.png
5
1
Hide problems
China Northern Mathematical Olympiad 2013 , Problem 5
Find all non-integers
x
x
x
such that
x
+
13
x
=
[
x
]
+
13
[
x
]
.
x+\frac{13}{x}=[x]+\frac{13}{[x]} .
x
+
x
13
=
[
x
]
+
[
x
]
13
.
where
[
x
]
[x]
[
x
]
mean the greatest integer
n
n
n
, where
n
≤
x
.
n\leq x.
n
≤
x
.
7
1
Hide problems
China Northern Mathematical Olympiad 2013 , Problem 7
Suppose that
{
a
n
}
\{a_n\}
{
a
n
}
is a sequence such that
a
n
+
1
=
(
1
+
k
n
)
a
n
+
1
a_{n+1}=(1+\frac{k}{n})a_{n}+1
a
n
+
1
=
(
1
+
n
k
)
a
n
+
1
with
a
1
=
1
a_{1}=1
a
1
=
1
.Find all positive integers
k
k
k
such that any
a
n
a_n
a
n
be integer.
2
1
Hide problems
China Northern Mathematical Olympiad 2013 , Problem 2
If
a
1
,
a
2
,
⋯
,
a
2013
∈
[
−
2
,
2
]
a_1,a_2,\cdots,a_{2013}\in[-2,2]
a
1
,
a
2
,
⋯
,
a
2013
∈
[
−
2
,
2
]
and
a
1
+
a
2
+
⋯
+
a
2013
=
0
a_1+a_2+\cdots+a_{2013}=0
a
1
+
a
2
+
⋯
+
a
2013
=
0
, find the maximum of
a
1
3
+
a
2
3
+
⋯
+
a
2013
3
a^3_1+a^3_2+\cdots+a^3_{2013}
a
1
3
+
a
2
3
+
⋯
+
a
2013
3
.