# S59 - Solution to Problem 59 Congratulations to this week's winner Nathan Pauli A partial solution was also received from Ray Kremer Further

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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.

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S59 - Solution to Problem 59 Congratulations to this week's winner Nathan Pauli A partial solution was also received from Ray Kremer Further

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