MathDB
Problems
Contests
National and Regional Contests
China Contests
China Northern MO
2006 China Northern MO
2006 China Northern MO
Part of
China Northern MO
Subcontests
(6)
8
1
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Sequence Inequality
Given a sequence
{
a
n
}
\{ a_{n}\}
{
a
n
}
such that
a
n
+
1
=
a
n
+
1
2006
a
n
2
a_{n+1}=a_{n}+\frac{1}{2006}a_{n}^{2}
a
n
+
1
=
a
n
+
2006
1
a
n
2
,
n
∈
N
n \in N
n
∈
N
,
a
0
=
1
2
a_{0}=\frac{1}{2}
a
0
=
2
1
. Prove that
1
−
1
2008
<
a
2006
<
1
1-\frac{1}{2008}< a_{2006}< 1
1
−
2008
1
<
a
2006
<
1
.
7
1
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64 grids
Can we put positive integers
1
,
2
,
3
,
⋯
64
1,2,3, \cdots 64
1
,
2
,
3
,
⋯
64
into
8
×
8
8 \times 8
8
×
8
grids such that the sum of the numbers in any
4
4
4
grids that have the form like
T
T
T
(
3
3
3
on top and
1
1
1
under the middle one on the top, this can be rotate to any direction) can be divided by
5
5
5
?
5
1
Hide problems
a+b+c = 3
a
,
b
,
c
a,b,c
a
,
b
,
c
are positive numbers such that
a
+
b
+
c
=
3
a+b+c=3
a
+
b
+
c
=
3
, show that:
a
2
+
9
2
a
2
+
(
b
+
c
)
2
+
b
2
+
9
2
b
2
+
(
a
+
c
)
2
+
c
2
+
9
2
c
2
+
(
a
+
b
)
2
≤
5
\frac{a^{2}+9}{2a^{2}+(b+c)^{2}}+\frac{b^{2}+9}{2b^{2}+(a+c)^{2}}+\frac{c^{2}+9}{2c^{2}+(a+b)^{2}}\leq 5
2
a
2
+
(
b
+
c
)
2
a
2
+
9
+
2
b
2
+
(
a
+
c
)
2
b
2
+
9
+
2
c
2
+
(
a
+
b
)
2
c
2
+
9
≤
5
4
1
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Range of Delta of Quadratic Function
Given a function
f
(
x
)
=
x
2
+
a
x
+
b
f(x)=x^{2}+ax+b
f
(
x
)
=
x
2
+
a
x
+
b
with
a
,
b
∈
R
a,b \in R
a
,
b
∈
R
, if there exists a real number
m
m
m
such that
∣
f
(
m
)
∣
≤
1
4
\left| f(m) \right| \leq \frac{1}{4}
∣
f
(
m
)
∣
≤
4
1
and
∣
f
(
m
+
1
)
∣
≤
1
4
\left| f(m+1) \right| \leq \frac{1}{4}
∣
f
(
m
+
1
)
∣
≤
4
1
, then find the maximum and minimum of the value of
Δ
=
a
2
−
4
b
\Delta=a^{2}-4b
Δ
=
a
2
−
4
b
.
2
1
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Sequence and Divisibility
p
p
p
is a prime number that is greater than
2
2
2
. Let
{
a
n
}
\{ a_{n}\}
{
a
n
}
be a sequence such that
n
a
n
+
1
=
(
n
+
1
)
a
n
−
(
p
2
)
4
na_{n+1}= (n+1) a_{n}-\left( \frac{p}{2}\right)^{4}
n
a
n
+
1
=
(
n
+
1
)
a
n
−
(
2
p
)
4
. Show that if
a
1
=
5
a_{1}=5
a
1
=
5
, the
16
∣
a
81
16 \mid a_{81}
16
∣
a
81
.
1
1
Hide problems
Prove midpoint
A
B
AB
A
B
is the diameter of circle
O
O
O
,
C
D
CD
C
D
is a non-diameter chord that is perpendicular to
A
B
AB
A
B
. Let
E
E
E
be the midpoint of
O
C
OC
OC
, connect
A
E
AE
A
E
and extend it to meet the circle at point
P
P
P
. Let
D
P
DP
D
P
and
B
C
BC
BC
meet at
F
F
F
. Prove that
F
F
F
is the midpoint of
B
C
BC
BC
.