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Solution to Problem 59
Congratulations to this week’s winner
Nathan Pauli
A partial solution was also received from Ray Kremer. Further correct solutions were also received
from William Webb, Robert McQuaid, Steven Young, Burkart Venzke. Two incorrect solutions were
submitted.
Many different solutions were submitted.
The one which follows contains ideas from
various solutions as well as some ideas of mine.
Start with the cubic
f(x) = ax
3
+ bx
2
+ cx + d
, where the coefficents
a,b,c,d
are all
integers.
In order for the stated conditions to hold, the roots of the first and second
derivative of
f(x)
must be integers, and the roots of the first derivative must be
distinct.
The second derivative is
f ’’(x) = 6ax + 2b
with the root
b/3a
; in particular,
3a
divides
b
.
Now consider the cubic polynomial
g(x)
=
f (x  b/3a)
which is also polynomial with
integer coefficients, and whose graph is the same as that of
f(x)
only shifted to the
right by
b/3a
units.
Therefore
f(x)
satisfies the desired conditions exactly when
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This note was uploaded on 03/01/2010 for the course MATH 301 taught by Professor Albertodelgado during the Spring '10 term at Bradley.
 Spring '10
 AlbertoDelgado
 Combinatorics

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