Tensor's Test

In the 1990s there used to be a tradition of newly joined IITians from Hyderabad, to conduct a pre-IIT JEE examination for the next batch, called the Tensor's test (is it still going on?). IIT-JEE =  joint entrance examination for admission to the IITs.

I had the opportunity to conduct the test (with Amit, Rajasekhar and Ramana).

Here are a few math questions from that test.

1) $f:\mathbb{R}\to\mathbb{R}$ is a continuous function such that $f(f(x)) = x,\ \forall x \in \mathbb{R}$. Show that, for some $c \in \mathbb{R}, f(c) = c$.

2) Find all natural numbers $x, y, x \gt y$, such that $x^y = y^x$. Hint (given as part of the test): What is the maximum value of $f(x) = x^{1/x}$?

3) $f_n$ is a sequence such that $f_1 \gt 0$ and $$3f_{n+1} = 2f_n + \frac{A}{f_n^2}$$ for some constant $A \gt 0$ and all $n \ge 1$. Show that $f_{n+1} \le f_n \forall n \gt 1$

Problem 3 was a particular favourite of mine (at that time) as I had discovered a way to compute $n^{th}$-roots (the above sequence converges to $\sqrt[3]{A}$), with a nice (in my opinion :-)) elementary proof that the sequence is bounded and monotonic.

Problem 3 basically asks for the monotonicity part of that proof. Only later did I realize that this was just Netwon-Raphson's method.

[If you are interested, the solutions to the above three problems are here: http://ruffnsluff.blogspot.com/p/tensors-test-solutions.html]