Configs in Algebra?

 We start with the following problem:



(2005 Czech-Polish-Slovak Olympiad) Let $n$ be a positive integer. Solve over $\mathbb{R}_{\ge 0}$ the system

$$\sum_{i  =1}^n x_i^i = n$$ $$\sum_{i = 1}^n ix_i = \frac{n(n+1)}{2}.$$

Okay. The first thing that we see is that $x_i = 1 \forall i$ works. Also I'd expect this to be the only solution; the equation feels very "restricted" in some sense. (I'm not sure how to formalise this intuition, so if anyone could help with that, that would be nice.)

Also, the equations just seem to look like AM-GM? I'm not sure how exactly, but maybe I've done enough AM-GM ineq such as 2012 IMO 2 so that $x_i^i$ and $ix_i$ look like AM-GM. (Is this how geometers see configs?)

Anyway, we'd be highly likely to find an AM-GM equality case from here. And equality holds for $x_i = 1$. So we're motivated (using Sharpness Principle, for example) to use AM-GM in the following way:

$$x_i^i + (i-1) \ge i \left(x_i^i\right)^{\frac{1}{i}} $$

And summing this over all $i$, 

$n + \frac{n(n+1)}{2} - n \ge \frac{n(n+1)}{2}$. But equality holds, so equality holds everywhere, so $x_i = 1$ is indeed always true. $\square$

(One more thing to note is how nicely this ties in with nonnegative reals specifically.)

Moral(s) of the story?

1. Config intuition might possibly exist in alg?
2. Sharpness principle OP
3. Trust your gut, unless the problem is from ToT :)


Signing off!!

(EDIT)

I forgot this really cool problem, it's related quite a lot to the above problem

Problem(2011 Mexico): Solve over $\mathbb{R}$:
\[a_1^2 + a_1 - 1 = a_2\]
\[ a_2^2 + a_2 - 1 = a_3\]
\[ \vdots \]
\[a_{n}^2 + a_n - 1 = a_1\]


(I wrote this article on 13 dec 2024; this is being migrated to this blog.)

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