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AIME Problems
2007 AIME Problems
14
Polynomial
Polynomial
Source: AIME II 2007 #14
March 29, 2007
algebra
polynomial
quadratics
function
probability
functional equation
AMC
Problem Statement
Let
f
(
x
)
f(x)
f
(
x
)
be a polynomial with real coefficients such that
f
(
0
)
=
1
,
f(0) = 1,
f
(
0
)
=
1
,
f
(
2
)
+
f
(
3
)
=
125
,
f(2)+f(3)=125,
f
(
2
)
+
f
(
3
)
=
125
,
and for all
x
x
x
,
f
(
x
)
f
(
2
x
2
)
=
f
(
2
x
3
+
x
)
.
f(x)f(2x^{2})=f(2x^{3}+x).
f
(
x
)
f
(
2
x
2
)
=
f
(
2
x
3
+
x
)
.
Find
f
(
5
)
.
f(5).
f
(
5
)
.
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