Let α∈(1,+∞) be a real number, and let P(x)∈R[x] be a monic polynomial with degree 24, such that (i) P(0)=1. (ii) P(x) has exactly 24 positive real roots that are all less than or equal to α. Show that |P(1)|≤(195)5(α−1)24.
Problem
Source: Vietnam TST 2024 P4
Tags: algebra, polynomial, inequalities
27.03.2024 12:15
28.03.2024 05:50
Let α∈(1,+∞) be a real number, and let P(x)∈R[x] be a monic polynomial with degree n, such that (i) P(0)=1. (ii) P(x) has exactly n positive real roots that are all less than or equal to α. Prove that |P(1)|≤max{[n−⌊n⋅W(1e)W(1e)+1⌋⌊n⋅W(1e)W(1e)+1⌋]⌊n⋅W(1e)W(1e)+1⌋;[n−⌈n⋅W(1e)W(1e)+1⌉⌈n⋅W(1e)W(1e)+1⌉]⌈n⋅W(1e)W(1e)+1⌉}(α−1)n(Lambert W function)
28.03.2024 13:30
I think it should be exactly the outcome of my proof method (although I did not do the computations in terms of W but t0...). (But note that one nice thing about my solution is that you don't need to know anything about the value of t0.)
28.03.2024 15:18
Tintarn wrote: I think it should be exactly the outcome of my proof method (although I did not do the computations in terms of W but t0...). (But note that one nice thing about my solution is that you don't need to know anything about the value of t0.) Of course, the general numbers above are based on available evidence. That evidence is similar to yours. In terms of timing, the proofs in the link below (a discussion forum on my country's Mathematics) precede yours [url]https://www.facebook.com/photo/?fbid=7830056027028570&set=gm.1351390282241765&idorvanity=243634219684049 [/url] However, I respect all proofs, including yours, so in my post, I won't include proofs. Thank you!