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Putnam
1999 Putnam
1
Putnam 1999 A1
Putnam 1999 A1
Source:
December 22, 2012
Putnam
algebra
polynomial
calculus
derivative
analytic geometry
graphing lines
Problem Statement
Find polynomials
f
(
x
)
f(x)
f
(
x
)
,
g
(
x
)
g(x)
g
(
x
)
, and
h
(
x
)
h(x)
h
(
x
)
, if they exist, such that for all
x
x
x
,
∣
f
(
x
)
∣
−
∣
g
(
x
)
∣
+
h
(
x
)
=
{
−
1
if
x
<
−
1
3
x
+
2
if
−
1
≤
x
≤
0
−
2
x
+
2
if
x
>
0.
|f(x)|-|g(x)|+h(x)=\begin{cases}-1 & \text{if }x<-1\\3x+2 &\text{if }-1\leq x\leq 0\\-2x+2 & \text{if }x>0.\end{cases}
∣
f
(
x
)
∣
−
∣
g
(
x
)
∣
+
h
(
x
)
=
⎩
⎨
⎧
−
1
3
x
+
2
−
2
x
+
2
if
x
<
−
1
if
−
1
≤
x
≤
0
if
x
>
0.
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