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Falls apart from the substitution S_n=1/T_n. In the case T=1 this yields: T_n= 2/(n^2-n+2).

Glad that you are planning pursuing mathematics further! I am sure I will run into you at some stage in the future.

+1.

Definitely not annoying, its good that you think about this sort of thing carefully! I suppose from a HS point of view that can be taken as the definition of pi, although there are issues with this: namely that things like arc length and hence circumference are not properly defined (some kind...

Your reasoning is essentially correct barb, although there are some unstated results that you used implicitly. (Why must there exist a stationary point which is a local minimum? And why must the poly attain a global minimum at such a local min?) I personally used an inductive approach.

Ditto.

Re: MX2 Integration Marathon There is almost certainly a way to do this using mx2 level tricks but sledgehammers are fun to wield.

Prove that for all positive integers n, the polynomial: P_n(x)=\sum_{k=0}^{2n}\frac{x^k}{k!} has no real roots.

Let a,b,c > 0 with abc=1. Prove that: \frac{a}{(a+1)(b+1)}+\frac{b}{(b+1)(c+1)}+\frac{c}{(c+1)(a+1)}\geq 3/4.

a) Find a condition for the cubic polynomial x^3+Ax+B to have three real roots (where A,B are real). Let R be the region bounded by the parabola x^2=4y and the line y=20. b) Using a) or otherwise, find the probability of being able to draw three distinct normals from a randomly selected point...

Let a,b,n be positive integers. Prove by induction or otherwise that if n divides a+b and n^2 divides a^2 + b^2, then n^m divides a^m + b^m for all positive integers m.

Chad O. Dude is on a date with a girl at an Italian restaurant. They are sharing a bowl of spaghetti which contains exactly n pieces. They eat by simultaneously choosing an end of a piece of spaghetti and sucking it into their mouths, then repeating this process. Find the probability of Chad...

Re: MX2 Integration Marathon I=\frac{\pi(3\log a-4)}{16a^5}. Consider the integrand over the complex plane with a branch cut on the non-negative imaginary axis. Then use the residue theorem with the contour consisting of an epsilon upper semicircle about 0, an R upper semicircle about 0...

I have read it, it isn't that good.

For the record, the paper is considerably easier than most olympiads.

Lols at the reactions.

Hahaha that is kind of amazing.

Final copy + solutions near ready carrot?

Formal onesies should definitely be a thing.

Sounds cool! Yeah I definitely meet some pretty interesting people doing what I do. Will be sure to encourage onesie purchasing if he doesn't already have one. (He can wear a penguin tux onesie to your future wedding.)