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Chile National Olympiad
2008 Chile National Olympiad
3
3
Part of
2008 Chile National Olympiad
Problems
(1)
functional equation with strictly increasing function and f((x+y)/2)
Source: Chile Finals 2008 L2 p3
10/5/2022
Determine all strictly increasing functions
f
:
R
→
R
f : R \to R
f
:
R
→
R
such that for all
x
≠
y
x \ne y
x
=
y
to hold
2
[
f
(
y
)
−
f
(
x
+
y
2
)
]
f
(
x
)
−
f
(
y
)
=
f
(
x
)
−
f
(
y
)
2
[
f
(
x
+
y
2
)
−
f
(
x
)
]
\frac{2\left[f(y)-f\left(\frac{x+y}{2}\right) \right]}{f(x)-f(y)}=\frac{f(x)-f(y)}{2\left[f\left(\frac{x+y}{2}\right)-f(x) \right]}
f
(
x
)
−
f
(
y
)
2
[
f
(
y
)
−
f
(
2
x
+
y
)
]
=
2
[
f
(
2
x
+
y
)
−
f
(
x
)
]
f
(
x
)
−
f
(
y
)
algebra
functional equation
function