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Problems
Contests
National and Regional Contests
China Contests
South East Mathematical Olympiad
2007 South East Mathematical Olympiad
2007 South East Mathematical Olympiad
Part of
South East Mathematical Olympiad
Subcontests
(4)
3
2
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[x] denotes the greatest integer not exceeding x
Let
a
i
=
m
i
n
{
k
+
i
k
∣
k
∈
N
∗
}
a_i=min\{ k+\dfrac{i}{k}|k \in N^*\}
a
i
=
min
{
k
+
k
i
∣
k
∈
N
∗
}
, determine the value of
S
n
2
=
[
a
1
]
+
[
a
2
]
+
⋯
+
[
a
n
2
]
S_{n^2}=[a_1]+[a_2]+\cdots +[a_{n^2}]
S
n
2
=
[
a
1
]
+
[
a
2
]
+
⋯
+
[
a
n
2
]
, where
n
≥
2
n\ge 2
n
≥
2
. (
[
x
]
[x]
[
x
]
denotes the greatest integer not exceeding x)
geometric sequence
Find all triples
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
satisfying the following conditions: (i)
a
,
b
,
c
a,b,c
a
,
b
,
c
are prime numbers, where
a
<
b
<
c
<
100
a<b<c<100
a
<
b
<
c
<
100
. (ii)
a
+
1
,
b
+
1
,
c
+
1
a+1,b+1,c+1
a
+
1
,
b
+
1
,
c
+
1
form a geometric sequence.
4
2
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sequence problem
A sequence of positive integers with
n
n
n
terms satisfies
∑
i
=
1
n
a
i
=
2007
\sum_{i=1}^{n} a_i=2007
∑
i
=
1
n
a
i
=
2007
. Find the least positive integer
n
n
n
such that there exist some consecutive terms in the sequence with their sum equal to
30
30
30
.
inequality abc=1
Let
a
a
a
,
b
b
b
,
c
c
c
be positive real numbers satisfying
a
b
c
=
1
abc=1
ab
c
=
1
. Prove that inequality
a
k
a
+
b
+
b
k
b
+
c
+
c
k
c
+
a
≥
3
2
\dfrac{a^k}{a+b}+ \dfrac{b^k}{b+c}+\dfrac{c^k}{c+a}\ge \dfrac{3}{2}
a
+
b
a
k
+
b
+
c
b
k
+
c
+
a
c
k
≥
2
3
holds for all integer
k
k
k
(
k
≥
2
k \ge 2
k
≥
2
).
2
2
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semicircle problem
A
B
AB
A
B
is the diameter of semicircle
O
O
O
.
C
C
C
,
D
D
D
are two arbitrary points on semicircle
O
O
O
. Point
P
P
P
lies on line
C
D
CD
C
D
such that line
P
B
PB
PB
is tangent to semicircle
O
O
O
at
B
B
B
. Line
P
O
PO
PO
intersects line
C
A
CA
C
A
,
A
D
AD
A
D
at point
E
E
E
,
F
F
F
respectively. Prove that
O
E
OE
OE
=
O
F
OF
OF
.
right-angle triangle ABC
In right-angle triangle
A
B
C
ABC
A
BC
,
∠
C
=
90
\angle C=90
∠
C
=
90
°, Point
D
D
D
is the midpoint of side
A
B
AB
A
B
. Points
M
M
M
and
C
C
C
lie on the same side of
A
B
AB
A
B
such that
M
B
⊥
A
B
MB\bot AB
MB
⊥
A
B
, line
M
D
MD
M
D
intersects side
A
C
AC
A
C
at
N
N
N
, line
M
C
MC
MC
intersects side
A
B
AB
A
B
at
E
E
E
. Show that
∠
D
B
N
=
∠
B
C
E
\angle DBN=\angle BCE
∠
D
BN
=
∠
BCE
.
1
2
Hide problems
cubic equation
Determine the number of real number
a
a
a
, such that for every
a
a
a
, equation
x
3
=
a
x
+
a
+
1
x^3=ax+a+1
x
3
=
a
x
+
a
+
1
has a root
x
0
x_0
x
0
satisfying following conditions: (a)
x
0
x_0
x
0
is an even integer; (b)
∣
x
0
∣
<
1000
|x_0|<1000
∣
x
0
∣
<
1000
.
function f(x)
Let
f
(
x
)
f(x)
f
(
x
)
be a function satisfying
f
(
x
+
1
)
−
f
(
x
)
=
2
x
+
1
(
x
∈
R
)
f(x+1)-f(x)=2x+1 (x \in \mathbb{R})
f
(
x
+
1
)
−
f
(
x
)
=
2
x
+
1
(
x
∈
R
)
.In addition,
∣
f
(
x
)
∣
≤
1
|f(x)|\le 1
∣
f
(
x
)
∣
≤
1
holds for
x
∈
[
0
,
1
]
x\in [0,1]
x
∈
[
0
,
1
]
. Prove that
∣
f
(
x
)
∣
≤
2
+
x
2
|f(x)|\le 2+x^2
∣
f
(
x
)
∣
≤
2
+
x
2
holds for
x
∈
R
x \in \mathbb{R}
x
∈
R
.