I encourage anyone who thinks these are easy high-school problems to try to solve some. They're published (including this year's) at https://www.imo-official.org/problems.aspx. They make my head spin.
- A 3Blue1Brown video on a particularly nice and unexpectedly difficult IMO problem (2011 IMO, Q2): https://www.youtube.com/watch?v=M64HUIJFTZM
-- And another similar one (though technically Putnam, not IMO): https://www.youtube.com/watch?v=OkmNXy7er84
- Timothy Gowers (Fields Medalist and IMO perfect scorer) solving this year’s IMO problems in “real time”:
x+y=1
xy=1
The incredible thing is the explanation uses almost all reasoning steps that I am familiar with from basic algebra, like factoring, quadratic formula, etc. But it just comes together so beautifully. It gives you the impression that if you thought about it long enough, surely you would have come up with the answer, which is obviously wrong, at least in my case.
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.
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.
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.
no
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.
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.