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Contests
National and Regional Contests
China Contests
China Team Selection Test
1988 China Team Selection Test
2
sum equals product in function
sum equals product in function
Source: China TST 1988, problem 2
June 27, 2005
function
algebra
functional equation
algebra unsolved
Problem Statement
Find all functions
f
:
Q
↦
C
f: \mathbb{Q} \mapsto \mathbb{C}
f
:
Q
↦
C
satisfying (i) For any
x
1
,
x
2
,
…
,
x
1988
∈
Q
x_1, x_2, \ldots, x_{1988} \in \mathbb{Q}
x
1
,
x
2
,
…
,
x
1988
∈
Q
,
f
(
x
1
+
x
2
+
…
+
x
1988
)
=
f
(
x
1
)
f
(
x
2
)
…
f
(
x
1988
)
f(x_{1} + x_{2} + \ldots + x_{1988}) = f(x_1)f(x_2) \ldots f(x_{1988})
f
(
x
1
+
x
2
+
…
+
x
1988
)
=
f
(
x
1
)
f
(
x
2
)
…
f
(
x
1988
)
. (ii)
f
(
1988
)
‾
f
(
x
)
=
f
(
1988
)
f
(
x
)
‾
\overline{f(1988)}f(x) = f(1988)\overline{f(x)}
f
(
1988
)
f
(
x
)
=
f
(
1988
)
f
(
x
)
for all
x
∈
Q
x \in \mathbb{Q}
x
∈
Q
.
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