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x^{2n}+a_{2n-1}x^{2n-1}+...+a_1x + a_0

Source: VI Soros Olympiad 1990-00 R3 10.6 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

May 29, 2024
algebrapolynomial

Problem Statement

A natural number nn is given. Find the longest interval of a real line such that for numbers taken arbitrarily from it a0a_0, a1a_1, a2a_2, ......, a2n−1a_{2n-1} the polynomial x2n+a2n−1x2n−1+...+a1x+a0x^{2n}+a_{2n-1}x^{2n-1}+...+a_1x + a_0 has no roots on the entire real axis. (The left and right ends of the interval do not belong to the interval.)
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