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JOM
JOM 2024
3
3
Part of
JOM 2024
Problems
(1)
f(x)/y^2 - f(y)/x^2 <= (1/x - 1/y)^2
Source: JOM 2024 P3
2/29/2024
Find all functions
f
:
R
+
→
R
+
f:\mathbb{R}^+\rightarrow\mathbb{R}^+
f
:
R
+
→
R
+
such that for all
x
,
y
∈
R
+
x, y\in\mathbb{R}^+
x
,
y
∈
R
+
,
f
(
x
)
y
2
−
f
(
y
)
x
2
≤
(
1
x
−
1
y
)
2
\frac{f(x)}{y^2} - \frac{f(y)}{x^2} \le \left(\frac{1}{x}-\frac{1}{y}\right)^2
y
2
f
(
x
)
−
x
2
f
(
y
)
≤
(
x
1
−
y
1
)
2
(
R
+
\mathbb{R}^+
R
+
denotes the set of positive real numbers.)(Proposed by Ivan Chan Guan Yu)
algebra
functional equation