MathDB
Problems
Contests
National and Regional Contests
China Contests
(China) National High School Mathematics League
2006 China Second Round Olympiad
5
5
Part of
2006 China Second Round Olympiad
Problems
(1)
2006 China Second Round Olympiad Test 1 #5
Source:
9/28/2014
Suppose
f
(
x
)
=
x
3
+
log
2
(
x
+
x
2
+
1
)
f(x) = x^3 + \log_2(x + \sqrt{x^2+1})
f
(
x
)
=
x
3
+
lo
g
2
(
x
+
x
2
+
1
)
. For any
a
,
b
∈
R
a,b \in \mathbb{R}
a
,
b
∈
R
, to satisfy
f
(
a
)
+
f
(
b
)
≥
0
f(a) + f(b) \ge 0
f
(
a
)
+
f
(
b
)
≥
0
, the condition
a
+
b
≥
0
a + b \ge 0
a
+
b
≥
0
is
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
necessary and sufficient
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
not necessary but sufficient
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
necessary but not sufficient
<span class='latex-bold'>(A)</span>\ \text{necessary and sufficient}\qquad<span class='latex-bold'>(B)</span>\ \text{not necessary but sufficient}\qquad<span class='latex-bold'>(C)</span>\ \text{necessary but not sufficient}\qquad
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
necessary and sufficient
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
not necessary but sufficient
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
necessary but not sufficient
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
neither necessary nor sufficient
<span class='latex-bold'>(D)</span>\ \text{neither necessary nor sufficient}\qquad
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
neither necessary nor sufficient
logarithms