MathDB
Mini interpolations give a big interpolation

Source: RMM Extralist 2021 A2

September 18, 2023
interpolationRMM Shortlistalgebrapolynomial

Problem Statement

Let nn be a positive integer and let x1,…,xn,y1,…,ynx_1,\ldots,x_n,y_1,\ldots,y_n be integers satisfying the following condition: the numbers x1,…,xnx_1,\ldots,x_n are pairwise distinct and for every positive integer mm there exists a polynomial PmP_m with integer coefficients such that Pm(xi)−yiP_m(x_i) - y_i, i=1,…,ni=1,\ldots,n, are all divisible by mm. Prove that there exists a polynomial PP with integer coefficients such that P(xi)=yiP(x_i) = y_i for all i=1,…,ni=1,\ldots,n.
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