I couldn't solve this problem by myself. The solution, however, is elementary but you must be a genius to come up with it within four hours.

What's this bar? Does X | Y mean that X is divisible by Y? But a^2 + b^2 is typically bigger than ab + 1. So it appears to be a narrow class of a, b for which the premise is true. Maybe we should start by characterizing it?

>>16704295
X|Y means X divides Y, i.e. Y=kX for some k.

>>16704292 (OP)
>but you must be a genius to come up with it within four hours.
Ehh it's more like 2 hours, you need time for problems 4 and 5. But yes it's crazy to solve this without knowing about vieta jumping beforehand.

>>16704292 (OP)
The solution requires the use of Vieta jumping which already is stretching the definition of "elementary solution".

Add to the fact that the person who solved it (Valentin vornicu) albeit was a genius but still had to know about the esoteric formula the vieta's formula.

I didn't knew it, hence I poured in nearly 15 days on the problem and even after that I solved it with a makeshift vieta formula for quadratics.

>>16704313
>The solution requires the use of Vieta jumping which already is stretching the definition of "elementary solution"
It isn't stretching anything, it's pretty clearly an elementary solution
>Add to the fact that the person who solved it (Valentin vornicu) albeit was a genius but still had to know about the esoteric formula the vieta's formula
Plenty of people solved it tf you talking about?

>>16704314
> Nobody of the six members of the Australian problem committee could solve it … it was [then] 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 jury finally had the courage to choose it as the last problem of the competition. Eleven students gave perfect solutions.
Funny enough, the Bulgarian guy who found the best solution wasn't even gold medalist.

>>16704325
> Current Romanian President solved this problem.
Lol

>>16704292 (OP)
What is this | signify here? Like some mod(x) = 0 operation?

>>16704292 (OP)
>some random number theory problem pulled out of the ass
>you stop giving a shit about this stuff when you do undergraduate abstract algebra and learn basic ring theory
>retards on /sci/ worship the IMO as some holy grail

>>16704295
Another berenstein bear.it has been other way before.

Read this. It may change your life. No need to thank me.
https://old.reddit.com/r/math/comments/3dt0oi/how_do_the_top_imo_contestants_become_so/ct8f1ks/

>>16704435
No, it's nothing at all like that.

>>16704299
mathematicians cant just write that in plain English instead of using retarded notation?

>>16704508
why would you keep spelling out divides all the time lol what a stupid idea. one plus two equals three

>>16704448
>In 2011, JMO #5 took me two hours. In 2012, the same problem took me 30 seconds
That's because you already knew the answer the second time around, dipshit.

He's basically right in general though, gitting gud is like 30% extreme fluency in the base subject matter (beyond what you'd probably acquire just taking a normal class), and 70% grinding thousands of problems to build your experience and your repertoire of tricks. t. putnam HM

>>16704560
Von Neumann literally said you just get used to it. I don’t get why people keep arguing about this.

>>16704292 (OP)
This problem is not difficult. You play around a bit with properties of quadratic equations/roots and derive a contradiction by assuming a+b is minimal.

>>16704295
>What's this bar mean?
Shit, I assumed the people here were a little more advanced than that. Have any of you even taken a single math course?

>>16704560
>t. putnam HM
What’s that?

>>16704584
Honorable mention. Top 75-ish basically (top 50 akshually but who's counting)

Can this be done with just a compass and straight edge?

>>16704448
Why would you believe whatever that tranny Evan Chen said? Is he still stuck in mit?

>>16704435
no, people really use it both ways. I expect some professor misremembers it and writes an entire book, then it snowballs from there. same in machine learning with test set and validation set; half of the books use them swtched up compared to the other half.

>>16704576
>Shit, I assumed the people here were a little more advanced than that
don't be trolling. it is ambiguous, sometimes the meaning is reversed.

>>16704619
the meaning is never reversed dipshit

>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.
Does this just mean that it's pretty clear that the students getting perfect scores were likely given the answers from outside help?

>>16705178
Nah, it's just highlights the difference between "contest" math and "research" math. If you're familiar with standard tricks and techniques that these problems tend to use then the solutions aren't insanely hard, but if you're a real-world mathematician where those tricks aren't generally useful they probably wouldn't jump to mind.

>>16705195
Those problems are like shitty point and click games where some puzzles are impossible to solve without being in the state of the mind of the person who made them.

>>16704313
Maybe troll post but anyway I know Vornicu because I went to the same highschool as him. He did not compete in 1988 but in 2001 and 2002 , you can check his wikipedia page. Nicusor Dan the current president of Romania did indeed solve it.

>>16705178
Meh, it’s hard to conclude anything because most of us aren’t qualified.
Good contest result is still the best predictor for a successful math career.

>>16704292 (OP)
(a^2+b^2)/(ab+1) = (b^2 -a^2)/(1-ab)

(a+bi)/(c+di) = (b-ai)/(d-ci)

>4dson

>>16704313
>The solution requires the use of Vieta jumping
A solution uses of Vieta jumping
I'll make a better solution

>>16704292 (OP)
ab+1 is a straight curve = constant = 1.

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

a^2 and b^2 are hyperbolic curve so the product is also an hyperbolic curve so it's perfectly square.

>>16706488
>a^2 and b^2 are hyperbolic curve so the summ is also an hyperbolic curve so it's perfectly square.
sorry, I wrote too fast : fix.

>>16704292 (OP)
it’s vacuously true. there is not such pair [math]a,b[/math].

>>16706560
a=b=1

>>16704292 (OP)
Uhh, chatgpt o4-mini solved this in a few seconds. Are mathematicians obsolete now?

>>16704292 (OP)
how is this useful in our daily life?

>>16706833
It gives you grant money

>>16706818
>chatbot managed to regurgitate the solution to a massively well-documented problem
W-wow

>>16704292 (OP)
wtf is a "perfect" square

>>16707166
A perfect square is just a square number which is an integer that is the result of multiplying an integer by itself. So, for example, 1 = 1*1, 4= 2*2, 9= 3*3, 16 = 4*4, etc. I'm not really sure why you couldn't just google search this.