Let $r>0$ be a real number. We call a monic polynomial with complex coefficients $r$-good if all of its roots have absolute value at most $r$. We call a monic polynomial with complex coefficients primordial if all of its coefficients have absolute value at most $1$. a) Prove that any $1$-good polynomial has a primordial multiple. b) If $r>1$, prove that there exists an $r$-good polynomial that does not have a primordial multiple. Proposed by Pranjal Srivastava
Problem
Source: India IMOTC 2024 Day 4 Problem 1
Tags: Polynomials, algebra, polynomial
math_comb01
31.05.2024 10:24
(a) Let the polynomial be $P = \prod (x- \alpha_i)$, consider the polynomial $Q = \prod (x^{2^{i-1}} - (\alpha_i)^{2^{i-1}})$ which clearly works.
(b) Let $d$ be an integer so that $r^d >2$ then $x^d-r^d$ does not have a primordial multiple, Suppose for the sake of contradiction that it has a primordial multiple, then let $P(x) = (a_n x^n + \cdots + a_0)(x^d-r^d)$ now compare coefficients and apply triangle inequality to get a contradiction, and we’re done
raju122
01.06.2024 13:19
This problem is also PLAGARIZED! From the ELMO shortlist!!! Dear Indian TST selection committee, PLEASE PLEASE PLEASE MAKE SURE TO CHECK THAT A PROBLEM HAS NOT EVER APPEARED IN CONTEST before adding it into the tests!
TST Day 3 P3 was from the DeuX Mathematics Olympiad 2021 Shortlist A5.
And this problem (TST Day 4 P1) was the ELMO 2018 shortlist A4.
Extremely pathetic to see that a solid 1/6th of the problems in India TST 2024 are plagarized, and who knows how many more. Check out the following (original) problem: https://artofproblemsolving.com/community/c6h1665474p10581495
Rg230403
01.06.2024 13:43
raju122 wrote: This problem is also PLAGARIZED! From the ELMO shortlist!!! Dear Indian TST selection committee, PLEASE PLEASE PLEASE MAKE SURE TO CHECK THAT A PROBLEM HAS NOT EVER APPEARED IN CONTEST before adding it into the tests!
TST Day 3 P3 was from the DeuX Mathematics Olympiad 2021 Shortlist A5.
And this problem (TST Day 4 P1) was the ELMO 2018 shortlist A4.
Extremely pathetic to see that a solid 1/6th of the problems in India TST 2024 are plagarized, and who knows how many more. Check out the following (original) problem: https://artofproblemsolving.com/community/c6h1665474p10581495 Hi, The problem was not "plagiarized", the idea is fairly natural and someone came up with it, a lot of people look at the problems and had not seen it before. Of course, the team tries it's best to find similar problems but it's not always possible to do that. ELMOSL is not the most common source and it's fairly easy to miss it... This is not any justification for missing it but please be more respectful and understand that there is a difference between plagiarizing and missing things (which happens in all contests, many problems even in the EGMO level and on the ISL have appeared in contests before and people don't go around claiming plagiarism... we also don't have the power of having the committees of the size of the IMO jury/IMO jury and they occasionally still miss similarities). I don't know what you mean by "who knows how many more"... Okay found the Deux problem, this is an unfortunate occurence too. But please do not take credit away from the ones who came up with the problem themselves as well, it's just that somebody else had also come up with a similar idea before because it's fairly natural. Remark: I just saw that you created this account just to attack the TSTs and make just these two posts, please don't stoop to such levels to just disrespect people. These are problems that they would be proud of and you seem to dismiss them as plagiarism.
IndoMathXdZ
01.06.2024 17:29
raju122 wrote: This problem is also PLAGARIZED! From the ELMO shortlist!!! Dear Indian TST selection committee, PLEASE PLEASE PLEASE MAKE SURE TO CHECK THAT A PROBLEM HAS NOT EVER APPEARED IN CONTEST before adding it into the tests!
TST Day 3 P3 was from the DeuX Mathematics Olympiad 2021 Shortlist A5.
And this problem (TST Day 4 P1) was the ELMO 2018 shortlist A4.
Extremely pathetic to see that a solid 1/6th of the problems in India TST 2024 are plagarized, and who knows how many more. Check out the following (original) problem: https://artofproblemsolving.com/community/c6h1665474p10581495 As somebody who also had my fair share of organizing contests and proposing problems (and DeuX as well in this case), I think there's no definite guarantee to avoid 100% plagiarism especially with how many problems and contests there are nowadays. It quite often happens that some problem statement are quite simple and natural to consider, and thus it's totally possible that a lot of people can come up with it (this happened to me around 3 or 4 times by now) or maybe some contest are quite obscure and thus the problem publicity is not as well known. As unfortunate as that might be, I think DeuX SL 2021 A5 is also quite natural of a statement for people to consider, and the jury might have missed that (because it's a mock contest in the first place) -- and that's totally fine. No reason to attack the hardworking organizers just because of this, just be considerate P.S. Kudos to the India IMOTC Team for the amazing problem quality though!
raju122
02.06.2024 08:59
Rg230403 wrote: raju122 wrote: This problem is also PLAGARIZED! From the ELMO shortlist!!! Dear Indian TST selection committee, PLEASE PLEASE PLEASE MAKE SURE TO CHECK THAT A PROBLEM HAS NOT EVER APPEARED IN CONTEST before adding it into the tests!
TST Day 3 P3 was from the DeuX Mathematics Olympiad 2021 Shortlist A5.
And this problem (TST Day 4 P1) was the ELMO 2018 shortlist A4.
Extremely pathetic to see that a solid 1/6th of the problems in India TST 2024 are plagarized, and who knows how many more. Check out the following (original) problem: https://artofproblemsolving.com/community/c6h1665474p10581495 Hi, The problem was not "plagiarized", the idea is fairly natural and someone came up with it, a lot of people look at the problems and had not seen it before. Of course, the team tries it's best to find similar problems but it's not always possible to do that. ELMOSL is not the most common source and it's fairly easy to miss it... This is not any justification for missing it but please be more respectful and understand that there is a difference between plagiarizing and missing things (which happens in all contests, many problems even in the EGMO level and on the ISL have appeared in contests before and people don't go around claiming plagiarism... we also don't have the power of having the committees of the size of the IMO jury/IMO jury and they occasionally still miss similarities). I don't know what you mean by "who knows how many more"... Okay found the Deux problem, this is an unfortunate occurence too. But please do not take credit away from the ones who came up with the problem themselves as well, it's just that somebody else had also come up with a similar idea before because it's fairly natural. Remark: I just saw that you created this account just to attack the TSTs and make just these two posts, please don't stoop to such levels to just disrespect people. These are problems that they would be proud of and you seem to dismiss them as plagiarism. Hello, Please note that the preceding post was NOT meant to discomfit or put down any of the Indian TST problem selectors. It was written, rather, to address the irresponsibility of the committee in commiting to choose problems for the test without adequately cross-checking whether they had previously appeared in contests. Indeed, both of the said problems stem from very natural ideas, but this only increases the probability of them having appeared in past contests, so an extra validation would have been required. Again, PLEASE PLEASE NOTE that no disrespect was intended, and apologies if any of the same was felt on the other side.
idkk
02.06.2024 13:22
math_comb01 wrote:
(a) Let the polynomial be $P = \prod (x- \alpha_i)$, consider the polynomial $Q = \prod (x^{2^{i-1}} - (\alpha_i)^{2^{i-1}})$ which clearly works.
(b) Let $d$ be an integer so that $r^d >2$ then $x^d-r^d$ does not have a primordial multiple, Suppose for the sake of contradiction that it has a primordial multiple, then let $P(x) = (a_n x^n + \cdots + a_0)(x^d-r^d)$ now compare coefficients and apply triangle inequality to get a contradiction, and we’re done
Can u explain ur part (a) in detail i essetntially did the same thing but i i considered some large number $n$ and replace with $x^n-(\alpha_i)^{n}$ and then worked out some more cases for when $|\alpha_i|$ is exactly $1$
idkk
02.06.2024 14:02
Btw here is an alternate construction for part b) . We show on a large enough choice of $t$ $P(x)=(x-r)^t(x-z_1)....(x-z_k)$ works. where $k$ is also large nough and $z_1,z_2,...,z_k$ are non zero complex nos satisying the modulus inequality obviously. Say F.T.S.O.C it has a primordial multiple $G(x)$ . Clearly r is a root of $G(x),G'(x),....,\frac{d^t}{dx^t} G(x)$ . Say $G(x)$ has degree $d$ . Suppose $G(x)=\sum_{i=0}^{d} a_ix^{i},a_d=1$ $\frac{d^t}{dx^t} G(x)=\sum_{i=0}^{d} a_i i(i-1)...(i-t+1)x^{i-t}$ We basically use $z_1,....z_k$ to keep $d$ as large as we want . By basic inequalites if we can show that $r^{d-t} d(d-1)....(d-t+1) > \sum_{i=0}^{d-1}r^{i-t} i(i-1)...(i-t+1) $ we are done $\iff d(d-1)....(d-t+1) > \sum _{i=t}^{d-1} r^{i-d} i(i-1)....(i-t+1)$ USe chebyshevs inequality to obtain that $\sum _{i=t}^{d-1} r^{i-d} i(i-1)....(i-t+1) \leq (1+\frac{1}{r}+\frac{1}{r^2}+....)\sum_{i=0}^{d-1} i(i-1)...(i-t+1) * \frac{1}{d-t}$ USe standard summation formulas to show it suffices to show $\frac{r}{r-1} \frac{1}{t} \leq 1$ which must happens for large enough $t$