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China National Olympiad
1990 China National Olympiad
3
3
Part of
1990 China National Olympiad
Problems
(1)
China Mathematical Olympiad 1990 problem3
Source: China Mathematical Olympiad 1990 problem3
10/18/2013
A function
f
(
x
)
f(x)
f
(
x
)
defined for
x
≥
0
x\ge 0
x
≥
0
satisfies the following conditions: i. for
x
,
y
≥
0
x,y\ge 0
x
,
y
≥
0
,
f
(
x
)
f
(
y
)
≤
x
2
f
(
y
/
2
)
+
y
2
f
(
x
/
2
)
f(x)f(y)\le x^2f(y/2)+y^2f(x/2)
f
(
x
)
f
(
y
)
≤
x
2
f
(
y
/2
)
+
y
2
f
(
x
/2
)
; ii. there exists a constant
M
M
M
(
M
>
0
M>0
M
>
0
), such that
∣
f
(
x
)
∣
≤
M
|f(x)|\le M
∣
f
(
x
)
∣
≤
M
when
0
≤
x
≤
1
0\le x\le 1
0
≤
x
≤
1
. Prove that
f
(
x
)
≤
x
2
f(x)\le x^2
f
(
x
)
≤
x
2
.
function
inequalities
induction
algebra unsolved
algebra