MathDB
Problems
Contests
National and Regional Contests
China Contests
China Northern MO
2010 China Northern MO
2010 China Northern MO
Part of
China Northern MO
Subcontests
(8)
7
1
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[x,y,z] =(x,y)+(y,z) + (z,x) with x <= y <= z and (x,y,z) = 1
Find all positive integers
x
,
y
,
z
x, y, z
x
,
y
,
z
that satisfy the conditions:
[
x
,
y
,
z
]
=
(
x
,
y
)
+
(
y
,
z
)
+
(
z
,
x
)
,
x
≤
y
≤
z
,
(
x
,
y
,
z
)
=
1
[x,y,z] =(x,y)+(y,z) + (z,x), x\le y\le z, (x,y,z) = 1
[
x
,
y
,
z
]
=
(
x
,
y
)
+
(
y
,
z
)
+
(
z
,
x
)
,
x
≤
y
≤
z
,
(
x
,
y
,
z
)
=
1
The symbols
[
m
,
n
]
[m,n]
[
m
,
n
]
and
(
m
,
n
)
(m,n)
(
m
,
n
)
respectively represent positive integers, the least common multiple and the greatest common divisor of
m
m
m
and
n
n
n
.
4
1
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chess pieces at intersections of 7x7 grid table
As shown in the figure, chess pieces are placed at the intersection points of the
64
64
64
grid lines of the
7
×
7
7\times 7
7
×
7
grid table. At most
1
1
1
piece is placed at each point, and a total of
k
k
k
left chess pieces are placed. No matter how they are placed, there will always be
4
4
4
chess pieces, and the grid in which they are located the points form the four vertices of a rectangle (the sides of the rectangle are parallel to the grid lines). Try to find the minimum value of
k
k
k
. https://cdn.artofproblemsolving.com/attachments/5/b/23a79f43d3f4c9aade1ba9eaa7a282c3b3b86f.png
3
1
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1 + 2^x 3^y=5^z , diophantine
Find all positive integer triples
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
such that
1
+
2
x
⋅
3
y
=
5
z
1 + 2^x \cdot 3^y=5^z
1
+
2
x
⋅
3
y
=
5
z
is true.
1
1
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a_n=2^{2n}a_{n-1}+n\cdot 2^{n^2}, a_1=2
It is known that the sequence
{
a
n
}
\{a_n\}
{
a
n
}
satisfies
a
1
=
2
a_1=2
a
1
=
2
,
a
n
=
2
2
n
a
n
−
1
+
n
⋅
2
n
2
a_n=2^{2n}a_{n-1}+n\cdot 2^{n^2}
a
n
=
2
2
n
a
n
−
1
+
n
⋅
2
n
2
,
(
n
≥
2
)
(n \ge 2)
(
n
≥
2
)
, find the general term of
a
n
a_n
a
n
.
6
1
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AK bisects BC, touchpoints of incircle related
Let
⊙
O
\odot O
⊙
O
be the inscribed circle of
△
A
B
C
\vartriangle ABC
△
A
BC
, with
D
D
D
,
E
E
E
,
N
N
N
the touchpoints with sides
A
B
AB
A
B
,
A
C
AC
A
C
,
B
C
BC
BC
respectively. Extension of
N
O
NO
NO
intersects segment
D
E
DE
D
E
at point
K
K
K
. Extension of
A
K
AK
A
K
intersects segment
B
C
BC
BC
at point
M
M
M
. Prove that
M
M
M
is the midpoint of
B
C
BC
BC
. https://cdn.artofproblemsolving.com/attachments/a/6/a503c500178551ddf9bdb1df0805ed22bc417d.png
2
1
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CE = EF wanted, tangents and secant from the same point
From a point
P
P
P
exterior of circle
⊙
O
\odot O
⊙
O
, we draw tangents
P
A
PA
P
A
,
P
B
PB
PB
and the secant
P
C
D
PCD
PC
D
. The line passing through point
C
C
C
parallel to
P
A
PA
P
A
intersects chords
A
B
AB
A
B
,
A
D
AD
A
D
at points
E
E
E
,
F
F
F
respectively. Prove that
C
E
=
E
F
CE = EF
CE
=
EF
. https://cdn.artofproblemsolving.com/attachments/8/c/bf15595bc341b917df30b3aa93067887317c65.png
8
1
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China Northern Mathematical Olympiad 2010 , Problem 8
Let
x
,
y
,
z
∈
[
0
,
1
]
x,y,z \in [0,1]
x
,
y
,
z
∈
[
0
,
1
]
, and
∣
y
−
z
∣
≤
1
2
,
∣
z
−
x
∣
≤
1
2
,
∣
x
−
y
∣
≤
1
2
|y-z|\leq \frac{1}{2},|z-x|\leq \frac{1}{2},|x-y|\leq \frac{1}{2}
∣
y
−
z
∣
≤
2
1
,
∣
z
−
x
∣
≤
2
1
,
∣
x
−
y
∣
≤
2
1
. Find the maximum and minimum value of
W
=
x
+
y
+
z
−
y
z
−
z
x
−
x
y
W=x+y+z-yz-zx-xy
W
=
x
+
y
+
z
−
yz
−
z
x
−
x
y
.
5
1
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China Northern Mathematical Olympiad 2010 , Problem 5
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers such that
(
a
+
2
b
)
(
b
+
2
c
)
=
9
(a+2b)(b+2c)=9
(
a
+
2
b
)
(
b
+
2
c
)
=
9
. Prove that
a
2
+
b
2
2
+
2
b
3
+
c
3
2
3
≥
3.
\sqrt{\frac{a^2+b^2}{2}}+2\sqrt[3]{\frac{b^3+c^3}{2}}\geq 3.
2
a
2
+
b
2
+
2
3
2
b
3
+
c
3
≥
3.