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MMATHS 2021, Problem 7: Two Pairs Beats A Straight

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October 31, 2021
YaleMMATHS

Problem Statement

Let Pk(x)=(x−k)(x−(k+1))P_k(x) = (x-k)(x-(k+1)). Kara picks four distinct polynomials from the set {P1(x),P2(x),P3(x),…,\{P_1(x), P_2(x), P_3(x), \ldots , P12(x)}P_{12}(x)\} and discovers that when she computes the six sums of pairs of chosen polynomials, exactly two of the sums have two (not necessarily distinct) integer roots! How many possible combinations of four polynomials could Kara have picked?
Proposed by Andrew Wu
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