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India IMO Training Camp
2012 India IMO Training Camp
3
Determine function given two inequalities
Determine function given two inequalities
Source:
July 23, 2012
function
inequalities
induction
algebra proposed
algebra
Problem Statement
Let
R
+
\mathbb{R}^{+}
R
+
denote the set of all positive real numbers. Find all functions
f
:
R
+
⟶
R
f:\mathbb{R}^{+}\longrightarrow \mathbb{R}
f
:
R
+
⟶
R
satisfying
f
(
x
)
+
f
(
y
)
≤
f
(
x
+
y
)
2
,
f
(
x
)
x
+
f
(
y
)
y
≥
f
(
x
+
y
)
x
+
y
,
f(x)+f(y)\le \frac{f(x+y)}{2}, \frac{f(x)}{x}+\frac{f(y)}{y}\ge \frac{f(x+y)}{x+y},
f
(
x
)
+
f
(
y
)
≤
2
f
(
x
+
y
)
,
x
f
(
x
)
+
y
f
(
y
)
≥
x
+
y
f
(
x
+
y
)
,
for all
x
,
y
∈
R
+
x, y\in \mathbb{R}^{+}
x
,
y
∈
R
+
.
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