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China Team Selection Test
2023 China Team Selection Test
P22
Functional equation
Functional equation
Source: 2023 China TST Problem 22
April 1, 2023
algebra
functional equation
function
Problem Statement
Find all functions
f
:
Z
→
Z
f:\mathbb {Z}\to\mathbb Z
f
:
Z
→
Z
, satisfy that for any integer
a
{a}
a
,
b
{b}
b
,
c
{c}
c
,
2
f
(
a
2
+
b
2
+
c
2
)
−
2
f
(
a
b
+
b
c
+
c
a
)
=
f
(
a
−
b
)
2
+
f
(
b
−
c
)
2
+
f
(
c
−
a
)
2
2f(a^2+b^2+c^2)-2f(ab+bc+ca)=f(a-b)^2+f(b-c)^2+f(c-a)^2
2
f
(
a
2
+
b
2
+
c
2
)
−
2
f
(
ab
+
b
c
+
c
a
)
=
f
(
a
−
b
)
2
+
f
(
b
−
c
)
2
+
f
(
c
−
a
)
2
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