Snail, I hope you are just trying to tease me. I'll join the game one more time, though I fear that readers are already bored.

   Thus n can only be either 1 or 2, easily seen to yield no more than the two solutions mentioned.

Sorry, not proved by your friends argument.

As I wrote before, it was my error; n can in fact be either 1, or 2, or 3, or 4, by that argument. This leaves us with checking all values of x from 1 to 99 - a "finite" task.

   ("The details are left to the reader.")

There's a challenge.
(I only claimed to have "a bit more than school maths".)

It does not seem too difficult to me, just boring. Hint: if the inequality holds for some n (> 1), it will also hold for n+1, since the left side is nonnegative and increases by a factor 10, whereas the right side only increases by a factor (1 + 2/n + 1/n²), which is less than 4. By induction, the assertion is proved.

Wrong again? Requiring non-school maths?