/sci/ - Science & Math

>>8576049
Wow I get a prize for doing your homework?
Ps : use induction you stupid sack of shit
ps+ : change your major

>>8576059
You won't get it faggot. Even Terry Tao failed. I call you out, prove it!

>>8576059
>This problem was submitted in 1988 by West Germany. None of the six members of the Australian problem committee could solve it. Two of the members were George and Esther Szekeres, both famous problem solvers and problem creators. Since it was a number-theoretic problem, it was sent to the four most renowned Australian number theorists. They were asked to work on it for six hours. None of them could solve it in this time. The problem committee submitted it to the jury of the XXIX IMO marked with a double asterisk, which meant a superhard problem, possibly too hard to pose. After a long discussion, the jury finally had the courage to choose it as the last problem of the competition. Eleven students gave perfect solutions.
>Among the eleven contestants who answered the problem correctly were Fields Medallist Ngo Bao Chau and Putnam Fellow Ravi Vakil. Among the many who scored 1 point out of 7 were Fields Medallist Terence Tao (who was, after all, only 13 years old) and Putnam Fellow Jordan Ellenberg, who had perfect scores in 1987 and 1989.
It seems that you are a bit arrogant and ignorant, you should leave this board.

>>8576080
Glad they are based anons on this board.

>t. Numberphile fag
I know you got it from there

>>8576066
I could prove it right now fag if only I knew how to use latex correctly.

>>8576049
biologist here, k=a′^2+b^2/a′b+1

and this is a contradiction, because the pair a′,b has a smaller sum than the pair a,b which we chose to be minimal. QED

>>8576090
>claims how to prove it
>dont know how to use LaTeX

pick one faggot

Let us prove the general case: if a,b,n∈Na,b,n∈N and
c:=an+bn(ab)n−1+1
c:=an+bn(ab)n−1+1
then c∈Nc∈N is an nnth power.

Proof: In order to show, it is clear that we need to express cc as c=tnc=tn for some t∈Nt∈N, isn't it?

So let us consider a=tα,b=tβa=tα,b=tβ. Because unless a,ba,b themselves are power some other t∈Nt∈N, then statement will not be true in general. We are given that a,ba,b are integers (WOLOG we have chosen as positive integers) and hence if cc is supposed to be as tntn then a,ba,b have to be some power of tt; Otherwise, contradiction will arrive.

So we have
c=an+bn(ab)n−1+1
c=an+bn(ab)n−1+1
i.e.c=(tα)n+(tβ)n(tα+β)n−1−1
i.e.c=(tα)n+(tβ)n(tα+β)n−1−1
i.e.c=tnβ(tnα−nβ)+1t(α+β)(n−1)+1
i.e.c=tnβ(tnα−nβ)+1t(α+β)(n−1)+1
Intuition says that we can choose β=1β=1. But that would be particular stage. So let β=β0β=β0. Now in order to make cc as integer, we must have
t(α+β)(n−1)+1∣tnα−nβ+1
t(α+β)(n−1)+1∣tnα−nβ+1
which is true iff
(α+β0)(n−1)∣(nα−nβ)
(α+β0)(n−1)∣(nα−nβ)
iff(α+β0)(n−1)s=nα−nβfor somes∈N
iff(α+β0)(n−1)s=nα−nβfor somes∈N
soα=−β0(2n−1)(n−1)s−n
soα=−β0(2n−1)(n−1)s−n
We choose ss in such a way that αα is in NN. For the sake of simplicity we take s=−1s=−1 then α=β0(2n−1)α=β0(2n−1). Thus we have a=tα=tβ0(2n−1),b=tβ0,β0∈N.a=tα=tβ0(2n−1),b=tβ0,β0∈N. And consequently c=tnβ0c=tnβ0. Since cc should be nnth power, so it follows that β0=1β0=1 and hence α=2n−1α=2n−1 so that c=tnc=tn. We are done.

Alright some guy above me apparently just proved it for all powers fml.
We know that for any a2 + b2 that there is a term c2 +2abcos(y) which is equal to it.
What is proposed here is that there is also a term n^2ab + n^2, where n is an integer, equal to a2 + b2
This means that if for any a and b there are variables n (is element N) c and y such that
n^2ab + n^2 = c^2 + 2abcos(y)
then the statement is true.
set n = c
n^2ab = 2abcos(y)
n^2 = 2cos(y)
sqrt(2cos(y) ranges from -sqrt2 to sqrt2 which contains the integer 1.
qed?
Idk if my proof works desu...

https://en.wikipedia.org/wiki/Vieta_jumping#Standard_Vieta_jumping

let a = b = 1 then (a^2 + b^2) / (ab + 1) = 1 = 1^2 Q.E.D

Guys, this is problem number 6 on one of the Putnam challenges. You're not going to solve it unless you're IQ > 165 or so.

>>8576080
you are Quoran, Right ?

>>8577075
i'm screaming

>>8576103

What the fuck, the text is all wrongly formatted for me. Also, what the fuck, how did you do it in 30 minutes, are you fucking crazy or what?

I want to fucking kill myself right now, sweet Jesus.

>>8576049
This can be solved with Vieta Jumping, OP. It's a really obscure technique that most people probably wouldn't think of... but once you know it, problems of these types fall like dominoes.

Anyway I know the method but can't be arsed typing out the latex.

>>8577245
im only 20 iq pts behind a man (einstein) who debunked the majority of physics by thinking about trains. And seeing as physics deals with true complexity and other science do not, its even more reasonable for a 140 to be able to trash your kind than it was for a 160 to trash physics ;)
if ur iq was above 140 u wouldnt be as butthurt about the concept of a 140 hyp grad lurking this beta barn..
so the logical conclusion is that ur iq is lower than 140 (possibly much lower)

if you met me in real life you would instantly be blown into a world where all that mattered was getting me to acknowledge your existence and even hopefully be your friend... But i would probably find you to be a desperately uninteresting person and want you to exit my awareness as fast as possible.
the fact that i dont lie about my iq is what gives me the edge necessary to call you an inferior little girl who couldnt understand basic physics, and BE RIGHT ABOUT IT.

honestly if i woke up one morning and for some reason my thinking capacity had devolved to your level, it would take me less than 5 min to run to the kitchen and slit my throat...
maybe your going to tell me u work at goldman..
but ud still be way inferior, as is anybody who projects their iq to be higher than it is..

maybe thats the point, inferiorboy ;)

>>8577399
are you a scriptwriter of the big bang theory?

>>8577434
i didn't give you permission to reply to me. listen, it was entertaining, first. hell, it was actually funny.

but now i'm getting the feeling that you actually think that you're on my level. from the day i was born, i was destined for success, you little shit. i was brought into this world by a software engineer and and a prominent actuary, both of norwegian descent. from the moment that the fucking curtains were raised, i was set to dominate the stem fields. and because i had guardians that actually cared about me, i flourished. pretty soon, i was placed into a special school of correspondence, specifically i.m gelfand's school for gifted children in new england. i not only reached my parents' expectations, but i passed them with flying colors. i had raw talent. i was fucking better. i was surrounded by 6th graders, many from hong kong, whom were smarter than half the posters in this fucking thread. and now, where am i?

in mit, getting my double bs in electrical engineering and physics, with a stanford-binet tested iq of 147 at age 17.

get this through your head: you are nothing. i am worth more than your entire goddamn family. i'm smarter than you, better looking than you, taller than you, wealthier than you, and more employable than you. while you type out another post to get cheap laughs, i'll be simultaneously working with the brightest minds in the world and fornicating with my beautiful girlfriend. it gets on my nerves when people pretend to be better than me.

know your place, you fucking vermin. never, ever reply to my posts, again.

>>8577600
u liek boland?

>>8577600
WOW rude!!???

>>8577617

i hate this problem
when i see it, i immediately know to use the vieta root formulae (or "vieta jumping" if you're gay) to solve it, and i know how any why it works
but i can't be sure if i actually know how to solve this problem, or if i am just confident because i've seen it before
if there was another problem like this, could i solve it?
would i only know to use vieta tricks because of this problem?
where does that put me, as a math major?

i hate this feeling

>>8577399
>>8577600
get over your IQ fixation fgt pls
protip: defend your weakness, not your strength

>>8576049
actual hint : use vieta jumping

>>8578356
DAE feel like learning stuff from others instead of deriving all of mathematics yourself is cheating?

>>8578431

Dependence is a sign of weakness.

>>8576049
>>8576080
fuck man, I don't even want to do the mental effort to try do understand these proofs, or the vieta stuff, or..
I'm fucked ;_;

>>8577617
>>8577640
>>8578365
thanks for the (You)s

a = b = 1
(1^2 + 1 ^2)/(1*1 + 1) = 2/2 = 1^2

>>8576059
/sci/ fucking exposed

>>8577265
I do lurk in Quora

>>8577399

Your trolling is unfunny.

>>8577600
>Age 17

all that and still underaged b&

>>8577600
Does your double bs stand for double bullshit?

>>8581903
but it worked