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India IMO Training Camp
2024 India IMOTC
13
13
Part of
2024 India IMOTC
Problems
(1)
Beware the trap!
Source: India IMOTC Practice Test 1 Problem 1
5/31/2024
Find all functions
f
:
R
ā
R
f:\mathbb R \to \mathbb R
f
:
R
ā
R
such that
x
f
(
x
f
(
y
)
+
y
f
(
x
)
)
=
x
2
f
(
y
)
+
y
f
(
x
)
2
,
xf(xf(y)+yf(x))= x^2f(y)+yf(x)^2,
x
f
(
x
f
(
y
)
+
y
f
(
x
))
=
x
2
f
(
y
)
+
y
f
(
x
)
2
,
for all real numbers
x
,
y
x,y
x
,
y
.Proposed by B.J. Venkatachala
functional equation
algebra
IMOTC