prove that if a^2+b^2/(ab+1) is integer, it's a perfect square, a,b integers

can you prove it?

I'm sure there are plenty of people on here capable of IMO problems. A bit backhanded not to state the source, admittedly...

Reply 2

SimonM

generalebriety

I'm sure there are plenty of people on here capable of IMO problems. A bit backhanded not to state the source, admittedly...

This was a phenomenally hard IMO problem, often cited as the hardest one around. Engel's spiel about it is pretty scary.

Nowadays though well prepared contestants will treat it quite comfortably with Vieta jumping. (I remember there being lots of comments about how a Q3 (I think) which used the same method was much easier because of the method being now well known)

Reply 3

schoolstudent

what do you mean by unnecessary? do you mean harsh?
i can't solve it...that's why it's been necessary to me to ask for help.
i had no idea it was a problem with that kind of reputation, I just plucked it out of a past paper...i liked it as it was one of the easier ones to state.

sorry about not stating the source...i'm not sure what was going through my head when I didn't.

Reply 4

schoolstudent

so what if it's an IMO problem anyway?

Reply 5

schoolstudent

the fact it's an IMO problem shows it is very possible to solve it within an hour-2 hours using high school methods (assuming you have enough experience, which I'd imagine some of the uni students here would), and that what you're trying to prove is definitely true, so it's not like it's an unsolved problem and you're looking for a proof which doesn't exist.

Just checked, and eleven students gave perfect solutions in the exam to this question
But "Nobody of the six members of the Australian problem committee could solve
it. Two of the members were Georges Szekeres and his wife, both famous problem
solvers and problem posers. Because it was a number theory 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."

those students must have been good

Reply 6

generalebriety

schoolstudent

the fact it's an IMO problem shows it is very possible to solve it within an hour-2 hours using high school methods

Kinda depends on whether it was a good IMO problem or not.

schoolstudent

(assuming you have enough experience, which I'd imagine some of the uni students here would)

To be fair, university doesn't prepare you for the IMO. Nor does the IMO prepare you for university. Other than the fact that they're both maths, they're not really very related.

Reply 7

ghostwalker

For some info on Vieta jumping, which coincidentally uses this particular problem as an example:

http://reflections.awesomemath.org/2007_5/vieta_jumping.pdf

The website itself may be of interest to those into mathematical competitions by the looks of it.

See http://reflections.awesomemath.org/archives.html

Reply 8

schoolstudent

thanks for the links. i'll take a look at them

does forming more and more quadratics out of the b^2-4ac's for each previous quadratic not give a solution eventually? (i haven't thought too much about this problem, i posted it here in the hope of getting some other people's thoughts on it)

start by forming a quadratic in a from the a^2+b^2/(ab+1)=k

what happens when you try this approach?

Reply 9

ghostwalker

schoolstudent

does forming more and more quadratics out of the b^2-4ac's for each previous quadratic not give a solution eventually?

start by forming a quadratic in a from the a^2+b^2/(ab+1)=k

what happens when you try this approach?

Try it and see what happens.

You're proposing an idea, and then expecting someone on here to investigate it for you and tell you what happens. Sorry, it don't work that way.

Reply 10

schoolstudent

ghostwalker

Try it and see what happens.

You're proposing an idea, and then expecting someone on here to investigate it for you and tell you what happens. Sorry, it don't work that way.

I'll do it, but it'll take me ages.
I don't expect it'd take some people on here nearly as long. In fact, I'd expect that, given the experience, ability and knowledge of some people on here, they'd be able to work out whether it'd work extremely quickly, without getting at all bogged down.

also, some posters in this thread are already familiar with this problem, so have probably considered the (fairly obvious) method I've suggested, meaning the only work they'd have to do would be to post yes/no or did/didn't work for me.