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min root of x^3 + ax - a^3 - 29 = 0 (I Soros Olympiad 1994-99 Round 2 11.3)

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May 26, 2024
algebrapolynomial

Problem Statement

For each non-negative aa, consider the equation x3+ax−a3−29=0.x^3 + ax - a^3 - 29 = 0. Let xox_o be the positive root of this equation. Prove that for all a>0a > 0 such a root exists. What is the smallest value of xox_o?
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