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Problems
Contests
National and Regional Contests
China Contests
China Northern MO
2011 China Northern MO
2011 China Northern MO
Part of
China Northern MO
Subcontests
(8)
8
1
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|1-|2-...|(n-1)-|n-x||...||=x
It is known that
n
n
n
is a positive integer, and the real number
x
x
x
satisfies
∣
1
−
∣
2
−
.
.
.
∣
(
n
−
1
)
−
∣
n
−
x
∣
∣
.
.
.
∣
∣
=
x
.
|1-|2-...|(n-1)-|n-x||...||=x.
∣1
−
∣2
−
...∣
(
n
−
1
)
−
∣
n
−
x
∣∣...∣∣
=
x
.
Find the value of
x
x
x
.
5
1
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Pythagorean triples containing 30
If the positive integers
a
,
b
,
c
a, b, c
a
,
b
,
c
satisfy
a
2
+
b
2
=
c
2
a^2+b^2=c^2
a
2
+
b
2
=
c
2
, then
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
is called a Pythagorean triple. Find all Pythagorean triples containing
30
30
30
.
4
1
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partition of 1-29
Assume the
n
n
n
sets
A
1
,
A
2
.
.
.
,
A
n
A_1, A_2..., A_n
A
1
,
A
2
...
,
A
n
are a partition of the set
A
=
{
1
,
2
,
.
.
.
,
29
}
A=\{1,2,...,29\}
A
=
{
1
,
2
,
...
,
29
}
, and the sum of any elements in
A
i
A_i
A
i
,
(
i
=
1
,
2
,
.
.
.
,
n
)
(i=1,2,...,n)
(
i
=
1
,
2
,
...
,
n
)
is not equal to
30
30
30
. Find the smallest possible value of
n
n
n
.
3
1
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1 + 2^x 7^y=z^2, diophantine
Find all positive integer solutions
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
of the equation
1
+
2
x
⋅
7
y
=
z
2
1 + 2^x \cdot 7^y=z^2
1
+
2
x
⋅
7
y
=
z
2
.
1
1
Hide problems
a_n =(\sqrt3 +\sqrt2)^{2n}, b_n=a_n +1/a_n
It is known that the general term
{
a
n
}
\{a_n\}
{
a
n
}
of the sequence is
a
n
=
(
3
+
2
)
2
n
a_n =(\sqrt3 +\sqrt2)^{2n}
a
n
=
(
3
+
2
)
2
n
(
n
∈
N
∗
n \in N*
n
∈
N
∗
), let
b
n
=
a
n
+
1
a
n
b_n= a_n +\frac{1}{a_n}
b
n
=
a
n
+
a
n
1
. (1) Find the recurrence relation between
b
n
+
2
b_{n+2}
b
n
+
2
,
b
n
+
1
b_{n+1}
b
n
+
1
,
b
n
b_n
b
n
. (2) Find the unit digit of the integer part of
a
2011
a_{2011}
a
2011
.
6
1
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AC tangent to (ABD), tangent and secant from the same point
As shown in figure, from a point
P
P
P
exterior of circle
⊙
O
\odot O
⊙
O
, we draw tangent
P
A
PA
P
A
and the secant
P
B
C
PBC
PBC
. Let
A
D
⊥
P
O
AD \perp PO
A
D
⊥
PO
Prove that
A
C
AC
A
C
is tangent to the circumcircle of
△
A
B
D
\vartriangle ABD
△
A
B
D
. https://cdn.artofproblemsolving.com/attachments/a/f/32da6d4626bb3592cec19a4cf0202121ba64db.png
2
1
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concurrency related to interior point of incircle connecting vertices
As shown in figure , the inscribed circle of
A
B
C
ABC
A
BC
is intersects
B
C
BC
BC
,
C
A
CA
C
A
,
A
B
AB
A
B
at points
D
D
D
,
E
E
E
,
F
F
F
, repectively, and
P
P
P
is a point inside the inscribed circle. The line segments
P
A
PA
P
A
,
P
B
PB
PB
and
P
C
PC
PC
intersect respectively the inscribed circle at points
X
X
X
,
Y
Y
Y
and
Z
Z
Z
. Prove that the three lines
X
D
XD
X
D
,
Y
E
YE
Y
E
and
Z
F
ZF
ZF
have a common point. https://cdn.artofproblemsolving.com/attachments/e/9/bbfb0394b9db7aa5fb1e9a869134f0bca372c1.png
7
1
Hide problems
2011-china-northern-mathematical-olympiad, p7
In
△
A
B
C
\triangle ABC
△
A
BC
, then
1
1
+
cos
2
A
+
cos
2
B
+
1
1
+
cos
2
B
+
cos
2
C
+
1
1
+
cos
2
C
+
cos
2
A
≤
2
\frac{1}{1+\cos^2 A+\cos^2 B}+\frac{1}{1+\cos^2 B+\cos^2 C}+\frac{1}{1+\cos^2 C+\cos^2 A}\le 2
1
+
cos
2
A
+
cos
2
B
1
+
1
+
cos
2
B
+
cos
2
C
1
+
1
+
cos
2
C
+
cos
2
A
1
≤
2