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P(x)=a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+...+a_1x+a_0

Source: 2022 Argentina OMA Finals L3 p6

March 25, 2024
algebrapolynomial

Problem Statement

For every positive integer nn, we consider the polynomial of real coefficients, of 2n+12n+1 terms, P(x)=a2nx2n+a2n−1x2n−1+...+a1x+a0P(x)=a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+...+a_1x+a_0 where all coefficients are real numbers satisfying 100≤ai≤101100 \le a_i \le 101 for 0≤i≤2n0 \le i \le 2n. Find the smallest possible value of nn such that the polynomial can have at least one real root.
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