Comment by Tainnor
4 days ago
This is slightly tedious to do by hand but there isn't really anything interesting going on in that problem - it's just solving a quadratic equation over the complex numbers.
4 days ago
This is slightly tedious to do by hand but there isn't really anything interesting going on in that problem - it's just solving a quadratic equation over the complex numbers.
That isn't much of an argument; nothing in math is truly interesting if you take that approach. exp(i\pi)+1=0 could be said to be dis-interesting because it is just rotation on the complex plane. But it is the opposite - it is interesting because it turned out to be rotation on the complex plane but approached from summing infinite series.
Similarly you can say that solving a quadratic over complex numbers is dis-interesting, but it is actually an interesting puzzle because it is trying its best to pretend it isn't a quadratic. In many ways succeeding, it isn't a quadratic - there is no "2" in it.
It's "not interesting" because no novel insight has to be used in order to solve this. It's immediately obvious how to solve it, just follow the textbook procedure.
This is distinct both from other typical IMO problems that I've seen and from research mathematics which usually do require some amount of creativity.
> exp(i\pi)+1=0
If your definition of "exp(i*theta)" is literally "rotation of the number 1 by theta degrees counterclockwise", then indeed what you quoted is a triviality and contains no nugget of insight (how could it?).
It becomes nontrivial when your definition of "exp" is any of the following:
- The everywhere absolutely convergent power series sum_{i=0}^\infty z^n/n!
- The unique function solving the IVP y'=y, y(0)=1
- The unique holomorphic extension of the real-valued exponential function to the complex numbers
Going from any of these definitions to "exp(i*\pi)+1=0" from scratch requires quite a bit of clever mathematics (such as proving convergence of the various series, comparing terms, deriving the values of sin and cos at pi from their power series representation, etc.). That's definitely not something that a motivated high schooler would be able to derive from scratch.
Those definitions of exp are all immediately obvious and nearly the definition of textbook, every university calculus course covers them. That is the issue with defining interesting as novel - nothing generally known is novel any more. And they don't require any special maths - sum_{i=0}^\infty z^n/n! is literally just multiplication and addition.
The long and short of it is it just isn't possible to tell someone that their problem isn't interesting. Interest isn't an inherent property of an equation, it is the state of mind of the person looking at the equation. And in this case the x+y/xy is a classic interesting puzzle despite (really because of) how well known the solution is.
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It is not interesting because there is no “real” solution (pun intended).
If you go to the complex plane, you are re-defining the plane. If you redefine the plane, then you can do anything. The puzzle is about confusing the observer who is expecting a solution in a certain dimension.
It's true that it might be unexpected that there is no real solution. I also wouldn't have intuited that from the problem statement itself.
However, it's not like you have to go out of your way to look for the complex numbers in some creative way. At some point while solving the quadratic equation you'll have to take the root of a negative number. So the only choice is to reach for the complex numbers, your hand is kinda forced.