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National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
2008 Swedish Mathematical Competition
3
3
Part of
2008 Swedish Mathematical Competition
Problems
(1)
f(x)+f(y) <=f(x+y) if f(x)/x is increasing for x>0
Source: 2008 Swedish Mathematical Competition p3
4/27/2021
The function
f
(
x
)
f(x)
f
(
x
)
has the property that
f
(
x
)
x
\frac{f(x)}{x}
x
f
(
x
)
is increasing for
x
>
0
x>0
x
>
0
. Show that
f
(
x
)
+
f
(
y
)
≤
f
(
x
+
y
)
,
for all
x
,
y
>
0
f(x)+f(y) \leq f(x+y) \qquad , \qquad \text{for all } x,y>0
f
(
x
)
+
f
(
y
)
≤
f
(
x
+
y
)
,
for all
x
,
y
>
0
algebra
inequalities