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alpha_i such that polynomial has all real/distinct roots

Source: 1990 Putnam B5

July 12, 2013
algebrapolynomialPutnamcollege contests

Problem Statement

Is there an infinite sequence a0,a1,a2,⋯ a_0, a_1, a_2, \cdots of nonzero real numbers such that for n=1,2,3,⋯ n = 1, 2, 3, \cdots the polynomial pn(x)=a0+a1x+a2x2+⋯+anxn p_n(x) = a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n has exactly nn distinct real roots?
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