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Consecutive roots and interpolation

Source: RMM Shortlist 2023 A2

February 29, 2024

interpolationalgebraPolynomialsRMM Shortlist

Problem Statement

Fix an integer

n \geq 2

and let

a_1, \ldots, a_n

be integers, where

a_1 = 1

. Let

f(x) = \sum_{m=1}^n a_mm^x.

Suppose that

f(x) = 0

for some

K

consecutive positive integer values of

x

. In terms of

n

, determine the maximum possible value of

K

.

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