MIT Integration Bee 2024 Finals Solutions

[@laplace1139]

One way to speed up P5 is to just use the generating function of the fibonacci numbers, which is pretty well known. It is 1/(1-x-x^2) and subbing x=1/4 gives 1/(1-(1/4)-(1/16))=16/11, and multiplying by 1/4 gives 4/11 (as we have (1/4)^(n+1) in the sum, not (1/4)^n). I would guess this is what the contestants used to evaluate the sum

[@leofigoboh1611]

Hello, are you using Obsidian to write?

[@danielrosado3213]

I use Microsoft whiteboard along with a drawing pad to write

[@leofigoboh1611]

@@danielrosado3213 Thanks!

[@calcul8er205]

Oooh I had a slightly different solution for Finals question 5. Let f_n(x)=(floor((2^n)x)-floor((2^n)x-1/4))/2^n and suppose that x has binary representation given by x=0.a_1a_2...a_na_(n+1)a_(n+1).... Then f_n(x)=1/2^n if a_(n+1)=a_(n+2)=0 or f_n(x)=0 otherwise. Suppose that the binary representation of x contains it's it first 2 consecutive 0's at the (N+1)th and (N+2)th decimal place, so that max f_n(x)=1/2^N. Then the integral can be expressed as sum_{N>=1} int_(A_N)1/2^N dx =sum_{N>=1}|A_N|/2^N where A_N ={x in (0,1): binary expansion of x has the first consecutive 0's occur in the (N+1)th and (N+2)th place}. Probabilistically, |A_N|=P_N=P(in a random sequence of 0's and 1's, the first 2 consecutive 0's occur in the (N+1)th and (N+2)th position). This sum can be evaluated by identifying that P_N satisfies the linear recurrence P_N=P_(N-1)/2+P_(N-2)/4, with initial conditions, P_0=1/4 and P_1=1/8.

[@danielrosado3213]

Some other matholy people told me about this binary representation. I love it!! There are so many different ways to look at problems like this (literally)

[@candy-coatedcloud]

I was unaware of the feynman trick for the second problem, Is there any other way to solve it, I have tried subtitutions and stuff but cannot get to an answer

[@danielrosado3213]

I have not found another method yet, but that doesn’t mean that none exist. Possibly there may be a series solution?

[@candy-coatedcloud]

@@danielrosado3213 No, I have no clue anymore...

[@mathalysisworld]

Thanks!!!

[@mathalysisworld]

Bro make a good thumbnail. Only then your video will blast because content is super.

[@finnhogan5525]

Very epic bro

[@yisahak]

Can any one tell me the best resources to prepare for this exam?

[@danielrosado3213]

The best resource is simply doing past problems, available through the website. Also, youtube.com/@Silver-cu5up?si=WJde_QfYq1xD1L64 this channel has a whole series on different methods needed for integration bees

[@yisahak]

@@danielrosado3213 thanks! that was very helpful

[@bharadwajkk6823]

Doing this before JEE feels like I'm battling a final boss where I learn new stuff constantly and patterns and just stuff....I LOVE IT

[@jatloe]

hii