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16
Math Prize 2010 Problem 16
Math Prize 2010 Problem 16
Source:
November 14, 2010
quadratics
function
analytic geometry
algebra
polynomial
Problem Statement
Let
P
P
P
be the quadratic function such that
P
(
0
)
=
7
P(0) = 7
P
(
0
)
=
7
,
P
(
1
)
=
10
P(1) = 10
P
(
1
)
=
10
, and
P
(
2
)
=
25
P(2) = 25
P
(
2
)
=
25
. If
a
a
a
,
b
b
b
, and
c
c
c
are integers such that every positive number
x
x
x
less than 1 satisfies
∑
n
=
0
∞
P
(
n
)
x
n
=
a
x
2
+
b
x
+
c
(
1
−
x
)
3
,
\sum_{n = 0}^\infty P(n) x^n = \frac{ax^2 + bx + c}{{(1 - x)}^3},
n
=
0
∑
∞
P
(
n
)
x
n
=
(
1
−
x
)
3
a
x
2
+
b
x
+
c
,
compute the ordered triple
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
.
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