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2021 IMO Shortlist
A8
A8
Part of
2021 IMO Shortlist
Problems
(1)
Clean and Memorable Functional Equation with a Conceptual Solution
Source: 2021 ISL A8
7/12/2022
Determine all functions
f
:
R
→
R
f: \mathbb{R} \rightarrow \mathbb{R}
f
:
R
→
R
that satisfy
(
f
(
a
)
−
f
(
b
)
)
(
f
(
b
)
−
f
(
c
)
)
(
f
(
c
)
−
f
(
a
)
)
=
f
(
a
b
2
+
b
c
2
+
c
a
2
)
−
f
(
a
2
b
+
b
2
c
+
c
2
a
)
(f(a)-f(b))(f(b)-f(c))(f(c)-f(a)) = f(ab^2+bc^2+ca^2) - f(a^2b+b^2c+c^2a)
(
f
(
a
)
−
f
(
b
))
(
f
(
b
)
−
f
(
c
))
(
f
(
c
)
−
f
(
a
))
=
f
(
a
b
2
+
b
c
2
+
c
a
2
)
−
f
(
a
2
b
+
b
2
c
+
c
2
a
)
for all real numbers
a
a
a
,
b
b
b
,
c
c
c
.Proposed by Ankan Bhattacharya, USA
algebra
functional equation
function