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Israel Team Selection Test
2023 Israel TST
P3
Function from integers to naturals
Function from integers to naturals
Source: 2023 Israel TST Test 1 P3
March 23, 2023
TST
functional equation
algebra
function
Problem Statement
Find all functions
f
:
Z
→
Z
>
0
f:\mathbb{Z}\to \mathbb{Z}_{>0}
f
:
Z
→
Z
>
0
for which
f
(
x
+
f
(
y
)
)
2
+
f
(
y
+
f
(
x
)
)
2
=
f
(
f
(
x
)
+
f
(
y
)
)
2
+
1
f(x+f(y))^2+f(y+f(x))^2=f(f(x)+f(y))^2+1
f
(
x
+
f
(
y
)
)
2
+
f
(
y
+
f
(
x
)
)
2
=
f
(
f
(
x
)
+
f
(
y
)
)
2
+
1
holds for any
x
,
y
∈
Z
x,y\in \mathbb{Z}
x
,
y
∈
Z
.
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