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2010 AMC 10BProblem 25P21-25~10 minPrint
PolynomialsDivisibility & FactorsGCD & LCMApply an identity: SFFT, sum of squares, difference of cubes, VietaBound the quantity above/below or estimate to pin it down

Let a>0a > 0 , and let P(x)P(x) be a polynomial with integer coefficients such that

P(1)=P(3)=P(5)=P(7)=aP(1) = P(3) = P(5) = P(7) = a , and
P(2)=P(4)=P(6)=P(8)=−aP(2) = P(4) = P(6) = P(8) = -a .

What is the smallest possible value of aa ?

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Problem © Mathematical Association of America (MAA), American Mathematics Competitions. Reproduced for non-commercial educational use. Solution and commentary are original to this site.