Side Story 1-2: Museum Curios

There's AM-GM. Pigeonhole. Invariance. Muirhead (to a certain extent). These are tactics and tools to swear by - use them well, and they will grant you powers beyond your wildest dreams. They will bring you places.

Then there are these theorems. Some of them are mere museum curios; I've never encountered them in any question thrown to me.

For someone who collects theorems like stamps, it is the grandest moment when a problem demands that you cockily brandish your new-found weapon of mass deduction. Especially if you've mustered the temperance to prove it yourself without consulting Google, and succeeded. More especially if you forged the masterpiece yourself. (Of course, mindlessly applying theorems is the antithesis of competitive maths.) Hence, I always find it pitiful that a few cool-looking tools end up unused, like some sort of Chekhov's Gun left hanging in the wall as the curtain falls.

Of course, all this is relative. Contests change all the time; tools could rise and fall in utility. Moreover some will say my being a mere tenderfoot in the Math Games makes me fancy rare, endangered beasts out of the standard wildlife. And I will have to grudgingly say that that's plausible.

Lagrange's Identity.
$$ \left(\sum_{k=1}^{n}a^2_{k}\right)  \left(\sum_{k=1}^{n}b^2_{k}\right) - \left( \sum_{k=1}^n a_k b_k \right)^2  = \sum_{i=1}^{n-1} \sum_{j=i+1}^n (a_i b_j - a_j b_i)^2 = \frac{1}{2} \sum_{i=1}^n \sum_{j=1}^n (a_i b_j - a_j b_i)^2  $$
Granted its similarity to the vaunted Cauchy-Schwarz Inequality, I thought this beautiful identity would figure more into algebra problems. Alas, to the best of my memory, I have never had to use it. Still, it is intuitive, and lends much insight into how things multiply out.

Fibonacci is a Square

Ira Gessel's Problem H-187:

A positive integer $n$ is a Fibonacci number if and only if either $5n^2+4$ or $5n^2-4$ is a square.

It is intuitive to surmise that this very beautiful statement has found applications in computer science. I've never had the joy of finding it to be the crux of some problem, though.


Beatty's theorem.

Given two positive irrational reals $\alpha$ and $\beta$ so that $1/\alpha + 1/\beta = 1$, the sets $\left\{ \left\lfloor \alpha\right\rfloor ,\left\lfloor 2\alpha\right\rfloor ,\left\lfloor 3\alpha\right\rfloor \ldots\right\}$ and $\left\{ \left\lfloor \mathbf{\beta}\right\rfloor ,\left\lfloor 2\beta\right\rfloor ,\left\lfloor 3\beta\right\rfloor \ldots\right\} $ form a partition of the real numbers. 

 Who would have thought? It bears a certain likeness with that problem of a walking person with an irrational footstep destined to fall into the single pothole of an otherwise smooth planet.

Routh's theorem.

$$A = \left[ ABC \right]\left( \frac{(xyz - 1)^2}{(xz + x + 1)(yx + y + 1)(zy + z + 1)}\right)$$

This should have been useful. I surmise that only pure chance was behind my never being able to use this gem of a theorem. There's little point in memorizing it, since mass points and area formulae could bring about the same result.

The Hermite Identity.
$$\sum_{k=0}^{n-1} \left\lfloor x+\frac{k}{n} \right\rfloor = \left\lfloor nx \right\rfloor$$

Bordering between "it makes sense" and "really?!", we have the intuitive Hermite identity.


Do comment if you've had similar experiences with some cool but impractical theorems.