IMO Shortlist 2004 N1: MAGICAL TAU CONSTRUCTION

Yeah. Like I think about one every like 40 or 50 games someone will comment on the fact that I'm from MIT. Okay, wait. Okay, I want A such that number of positive divisors Is that tau?

number of positive divisors Is that tau? I thought you were doing math not talking about StarCraft. I'm just responding to the chat. I apologize, but let's do the N1. A equals 1. So the problem is A equals 1 doesn't even work cuz you can take like N equals

So the problem is A equals 1 doesn't even work cuz you can take like N equals 1 right? A equals 1 Like tau of 1 is 1. Am I crazy? Or N equals 2, yeah, also. Hi Evan Chen, hello.

Hi Evan Chen, hello. I appreciate the sentiment which is that there should be some size issues. Let me think this through. Yeah, there should be some size issues

Yeah, there should be some size issues though in some sense. Like A equals 1 hints at that. I bet A equals 2 doesn't work either. Like I almost think you should think of it as tau of M equals M over A for all A divided for some

divided for some M divisible by A. So is there a number such that is there an even number such that half the numbers below it are its divisors? Probably yes. I think N equals , is there any? Does A equals 2

, is there any? Does A equals 2 work? the first problem from today was 2000 2062 for the person in the chat that asked. N equals 8. 1 2 4 N equals 8 doesn't quite work, right?

N equals 8 doesn't quite work, right? I'm I'm a little confused. N equals 6. 1 2 3 6. No, one That doesn't work either, right? , and okay, N equals 6, so M equals 12. 1 2 3 4 6 12, yeah.

12. 1 2 3 4 6 12, yeah. How do you type LaTeX so fast? I have a lot of practice with this. I've been using LaTeX since I was 13, and I've been live typing LaTeX for like classes for like 4 or 5 years now.

classes for like 4 or 5 years now. Yeah, so like A equals 2 doesn't work, but I feel like something's off here, because What if I take like

A equals 4? Tau does not include N. Did I misread the problem? No, tau is number of divisors, so it should include N, right? Yeah. No, sorry. I

No, sorry. I'm sorry, that doesn't make sense. Yeah, size is going the wrong way,

. If you pick an A that's Well, let me let me make sure I don't mess this up. So, if I pick an A that's We shouldn't we find an example that works? I'm trying.

works? I'm trying. I don't see one yet. Does A equals 4 works?

Does A equals 4 works? Tau n is less than 2 root n. Is that true? That is true. Tau n is less than 2 root n. But that won't be good enough because wait really?

wait really? Wait, hang on. Tau n is less The number of divisors is always at most the square root of n. , that's a good point. Yeah, that's true. So, besides my carrier is So, the idea is

So, the idea is tau of n is less than or equal to square root a n. And my hope is that if this is strictly less than n So for

So for any given value of a, like there are only finitely many work. No solutions where n is greater than something. I guess I have to square both sides, right? 4 a squared. Evan, a cannot be prime because then a

has eight divisors. Okay, I see. Okay, so a That's

Okay, I see. Okay, so a That's worth noting. a equals prime never works. I'll say odd prime. Is there an greater than 4A? . yes, n greater than 4A. , my point with

n greater than 4A. , my point with this is that there's if I pick a value of A, then there will be no solutions when n is greater than 4A. So, if n is greater than 4A, then tau n is into this is strictly less than n.

, honestly, the two less than root n thing should be strict. Well, there's no whatever. No solution. Yeah, so what this means is that I just need to pick A to dodge finitely

many smaller cases. unfortunately, the number of smaller cases, obviously, is growing in terms of A. I'm really sad that all the odd primes fell though. Why doesn't A equals two work?

What was the counterexample? . , does n equals eight work for every prime? No, it's n equals four, yeah.

One, two, four, eight. Yeah, yeah. shoot. Stop spamming symbols. Okay, I need to turn that off.

Stop spamming symbols. Okay, I need to turn that off. There's going to be a lot of symbols in the math chat. Moderator view, Yeah, I think these moderating features

Yeah, I think these moderating features are a little I'm going to disable excess symbols.

I'm going to disable excess symbols. Right. What is up the world? Can we prove a equals four works?

I don't know what the status on a equals four was. I don't think we decided on that. A equals four fails for n equals nine.

A equals four fails for n equals nine. So, it looks like We might need to be really opportunistic with the A. Cuz it looks like most of the ones we're trying to keep having counter examples. This really bugs me because it's like for any given A, I can show

it's like for any given A, I can show that there's at most finitely many bad n. But, I want them to have no solutions at all. What am I missing? Tau of m divides m.

Tau of m divides m. I don't like this. I'm going to do an actual. So, if a times n Well, let me pick one.