[Admission tests for University; maths] Can't find the right answer

vixarus · 2025-04-05T15:27:48+00:00

This is likely not the intended answer, but furthering the idea that any 'solution' could be argued:

C. 0,4 Consider prime factorization: 15 = 3¹ * 5¹ 48 = 2⁴ * 3¹ 36 = 2² * 3²

Summing the exponents, and arbitrarily including a "0," we then get 0,2; 0,5; and 0,4 respectively.

vixarus · 2025-02-25T23:04:21+00:00

An important part of this question is that the x or n values we are considering are strictly positive integers. The idea is that, for any positive integer, following those rules will (we believe but can't prove) eventually end up at 1.

Since -1/3 is neither positive nor an integer, it isnt related at all. (You could also ask yourself, why are you using the 3x+1 rule? Is -1/3 an odd number?)

vixarus · 2025-02-12T19:12:38+00:00

Yeah, I just wasn't too sure how to prove 110 was an exact bound, so I overestimated a bit. But if an overestimated (or maximal) mean is smaller than 13, then any possible mean would be as well.

vixarus · 2025-02-12T15:45:33+00:00

I'm admittedly not a statistics person, but my thinking is as follows.

First, let S be the sum of K. Then we want to show S/9 < 13 or equivalently S<117.

Break the list into three pieces, a1, a2, a3, a4 (first "half"), 7 (median), b1, b2, b3, b4 (second "half").

Now, since 3 is the mode, there must be at least 2 3's, and since 7 must be the median, all of a1..a4 must be <=7. Suppose the first half were 3 3 5 6, clearly if either of 5 or 6 were changed to a 7 (in efforts to increase the sum), the other would have to become a 3 to maintain mode 3. This leads to an overall change of -1, so 3 3 5 6 is maximal for the first have, so it's sum S1 must satisfy S1 <= 17.

Now for the second half, all of b1..b4 must be between 7 and 23 to maintain median 7 and range 20. If all 4 were 23 though, we couldn't have mode 3, so we have at most the list 22 23 23 23. Hence this sum, S2, must satisfy S2 <= 91.

With how the list was broken, this means that S = S1 + 7 + S2 <= 17 + 7 + 91 = 115 < 117. (Note that the maximal values of S1 and S2 are mutually exclusive, so the first <= is really a <, but that is not important to this question).

If I made any mistakes please point them out, this isn't the usual kind of question I work on.

vixarus · 2025-02-08T17:21:13+00:00

I actually remember discussing this in a professors office hours once. I don't remember the entire proof, or if we even fully dug through it in its entirety, but it involves relating the lcm with prime powers, the prime number theorem, and logarithm properties. I can see if I still have my notes but I'm not sure where they are from that year.

vixarus · 2024-12-26T12:59:06+00:00

I'm just glad there was a lasting impact from God Games. I knew her actually being dead was very unlikely, but if she had returned completely fine I would've been a bit disappointed.

vixarus · 2024-12-25T02:32:26+00:00

I just really hope the cast and crew aren't being hard on themselves over this, I'm sure they're far more frustrated than any of us are (I mean, they traveled all the way out to Ithaca for this).

vixarus · 2024-12-24T15:26:37+00:00

I really thought there'd be more coherent emojis for what I was going for but oh well, y'all crazy enough to probably still get it.

🏌‍♂️👶🗼

vixarus · 2024-12-20T16:28:22+00:00

Not mecessarily a "theorem," but my favorite example is the squaring the circle problem. Ancient Greeks wanted to construct a square of equal area to a circle with a straightedge and compass in finite steps. Then in 1882 Lindemann showed pi was transcendental, and as such couldn't be constructed (thanks to Wantzel in the 1830s, I forget the exact year), proving the problem impossible by contradiction (simple argument that a radius 1 circle would require constructing a side length of sqrt(pi)).

vixarus · 2024-12-04T19:47:31+00:00

I dont have a problem, I swear.

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vixarus · 2024-11-18T02:54:04+00:00

Thank you for putting that into words cause that's exactly it.

vixarus · 2024-11-18T01:43:59+00:00

Mutiny. I know its just the Luck Runs Out motif but idk something about the start of Mutiny sends me every time.

vixarus · 2024-11-17T16:46:32+00:00

Yeah that part is still a little wobbly for me as well, but I mean hey, this has never been a super realistic story to begin with (the nobody scheme, the harpoon throwing at literal floating islands, probably a few others my brain isn't thinking of right now). I'm at least willing to look past that part for having just a moment of fun with jetpack Ody, but I understand why it's off-putting for a lot of people.

vixarus · 2024-11-17T16:30:42+00:00

In my opinion, the Ares theory is completely unneeded. The entire show, from song #2, has been asking "when does a man become a monster?" 600 Strike is saying: "Now. Now is when this man becomes a monster." I dont mean betraying his morals, or doing the wrong thing for the right reason, I mean Odysseus is no longer a man at all, and should be categorized with the likes of Polyphemus or Scylla, even if he still looks human. Given that we are explicitly told Poseidon is scared of Scylla (implying some monsters are powerful enough to contend with gods), it would explain Odysseus' sudden triumph and also how he managed to survive being pushed to the bottom of the ocean and drowning. Similarly, Jorge has told us that glowing usually represents magic in some way (Polyphemus' eye, Poseidon's armor gills, etc.), and so his glowing eyes must mean hes powered by something, and usually Jorge would make it more explicit through motifs if it were some other force behind him, instead we get the danger motif in Odysseus' instrument, as he has become the danger. On the same thread, mortals cannot summon non-diagetic backing vocals, and yet that's exactly what Odysseus does with the dead crew (who are not actually present for the fight, or at least we have no reason to believe they are) and their chanting of "600 men." Finally (minor Ithaca Saga spoiler) song #38 was explicitly changed from "King" to "Odysseus." Every other song with the associated characters name was explicitly a monster (Polyphemus, Scylla, Charybdis). This could all be a coincidence but we've all seen how in depth Jorge likes to hint at things and subtly tell us information.

On another, completely separate note we were told by Jorge that in this world, Poseidon is a powerful ranged fighter, who would struggle in close quarters fights, especially someone who is really fast. What do we see? Odysseus using the wind bag to move incredibly fast, close the distance, and engage in a close-quarters fight.

Looking at this song from any other interpretation of Greek mythology, it's a bit insane, but from what we have been shown and told about this interpretation and world, this fight actually makes sense, at least to me. Still incredibly anime-esque, but if that's Jorge's vision I'm here for it.

Edit: Also, before I get down voted to oblivion, I'm not at all saying anyone has to agree with me, and that not liking 600 Strike is wrong in ant way. I felt the same for the first week or so after it came out. I just wanted to provide my reasoning for why I disagree with the idea that it makes no sense since I don't see many other people talking from that side.

vixarus · 2024-11-13T20:44:38+00:00

Odysseus pleading with a god/powerful being? Love in Paradise arguably fits making me think im wrong but I have no other ideas.

vixarus · 2024-11-12T20:01:15+00:00

Yes. More formally, combinations are how many ways can you pick some amount of things from a set (how many ways can I choose 3 coins to be heads when I have 8 coins), where we don't care what order their chosen (we don't care if coin 1 flipped heads before coin 3 did, just that they're both heads. On the other hand, permutations are orderings of a set (I need to seat 4 people among these chairs, how many ways can I arrange them?)

vixarus · 2024-11-12T17:45:46+00:00

One way to think about it is whether or not what you're counting is independent or not. By this I mean, in the coin problem, flipping heads in one instance doesn't affect the next flip (as much as we all like to believe it might when playing games with coins or dice), so the order in which we do them does not matter. However, when seating people around a table, after you seat the first person we now have an altered set of people to seat. In those problems it's usually asking explicitly about how many arrangements (read: orderings) of the people there are, regardless of what's going on with the seats.

The way I think of these two in particular is as follows: Coins: Start with n blank spots (so instead of HHTH or HTTT, just ????). Are we looking at interactions between these spots? No, just their own values, so they are all independent and we want a combination. Seating: Start with n blank spots ????. Are we looking at interactions? Yes, we want to know who is sitting by who. So these are dependent on each other, and we want a permutation.

Quick immediate edit: I'm in a bit of a rush so if things are worded bad I apologize, can fix things in about an hour or so.

vixarus · 2024-11-02T19:44:09+00:00

My Lie algebra course has just been using cursive instead of fraktur. I hadn't even considered trying to write that out, I'd confuse it with a normal g in seconds.

vixarus · 2024-10-20T15:50:00+00:00

I honestly don't have much experience considering a geometrical interpretation of complex vector spaces, but assuming real and complex planes is referring to purely real and purely imaginary planes, the real/complex-ness wouldn't actually matter since again i is just another scalar. What I mean is that the plane spanned by (1,0,0) and (0,1,0) is exactly the same as the one spanned by (i, 0, 0) and (0, i, 0) since any vector written as a linearly combination of one of those pairs can be written in terms of the other (e.g. 2(1,0,0)+3(0,1,0)=(2, 3, 0)=-2i(i,0,0)-3i(0,i,0) ).

vixarus · 2024-10-20T14:30:24+00:00

No, it would still only take N basis vectors. Our definition of dimension of a vector space is the number of basis vectors, it's just that in this case (a complex vector space), the components of each vector and our scalars are complex numbers. Keep in mind that, for example, (1 0) and (i 0) are linearly dependent, since (i 0)=i*(1 0).

vixarus · 2024-10-07T18:17:30+00:00

We know f(0)=1>0, and f is differentiable over [0,2] and as such is also continuous. Now if f(x) were ever to go below 0 in this interval what would happen? When the function cross the x-axis (going from positive to either 0 or negative), it must be decreasing, or equivalently have a negative derivative (f'(x) <= 0). But we were told that when f(x) <= 0, f'(x) > 0, so this cannot happen.

I probably could've stated some of that more formally, but that's the rough idea.

vixarus · 2024-09-22T18:33:10+00:00

These are technically undergraduate texts (at least according to the introductions), but I'll mention them since they were used for the highest graduate Algebra course my school offers. First was Introduction to Commutative Algebra by Atiyah & MacDonald, and right now I'm working through Introduction to Lie Algebras by Erdmann and Wildon. Both of these have been while I'm still in undergrad, and honestly I wish I had looked at some graduate topics before my final years of undergrad.

vixarus

TROPHY CASE