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A weird polynomial inequality

Source: Vietnam TST 2024 P4

March 27, 2024
algebrapolynomialinequalities

Problem Statement

Let α∈(1,+∞)\alpha \in (1, +\infty) be a real number, and let P(x)∈R[x]P(x) \in \mathbb{R}[x] be a monic polynomial with degree 2424, such that
(i) P(0)=1P(0) = 1. (ii) P(x)P(x) has exactly 2424 positive real roots that are all less than or equal to α\alpha.
Show that ∣P(1)∣≤(195)5(α−1)24|P(1)| \le \left( \frac{19}{5}\right)^5 (\alpha-1)^{24}.
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