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3
2004 General, part 1 #3
2004 General, part 1 #3
Source:
March 8, 2024
algebra
Problem Statement
Suppose
f
f
f
is a function that assigns to each real number
x
x
x
a value
f
(
x
)
f(x)
f
(
x
)
, and suppose the equation
f
(
x
1
+
x
2
+
x
3
+
x
4
+
x
5
)
=
f
(
x
1
)
+
f
(
x
2
)
+
f
(
x
3
)
+
f
(
x
4
)
+
f
(
x
5
)
−
8
f(x_1 + x_2 + x_3 + x_4 + x_5) = f(x_1) + f(x_2) + f(x_3) + f(x_4) + f(x_5) - 8
f
(
x
1
+
x
2
+
x
3
+
x
4
+
x
5
)
=
f
(
x
1
)
+
f
(
x
2
)
+
f
(
x
3
)
+
f
(
x
4
)
+
f
(
x
5
)
−
8
holds for all real numbers
x
1
,
x
2
,
x
3
,
x
4
,
x
5
x_1, x_2,x_3, x_4, x_5
x
1
,
x
2
,
x
3
,
x
4
,
x
5
. What is
f
(
0
)
f(0)
f
(
0
)
?
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