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Israel Team Selection Test
2021 Israel TST
2
Variant on old FE, f(xf(x+y))
Variant on old FE, f(xf(x+y))
Source: 2021 Israel TST 8 P2
May 31, 2022
Problem Statement
Find all functions
f
:
R
→
R
f:\mathbb{R}\to \mathbb{R}
f
:
R
→
R
so that for any reals
x
,
y
x,y
x
,
y
the following holds:
f
(
x
⋅
f
(
x
+
y
)
)
+
f
(
f
(
y
)
⋅
f
(
x
+
y
)
)
=
(
x
+
y
)
2
f(x\cdot f(x+y))+f(f(y)\cdot f(x+y))=(x+y)^2
f
(
x
⋅
f
(
x
+
y
))
+
f
(
f
(
y
)
⋅
f
(
x
+
y
))
=
(
x
+
y
)
2
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