How are inequalities from IMO built?

Solution 1:

Inequalities are very easy to compose. You start from self-evident inequalities like $x^2\geq0$ or well know ones like inequalities between means, or between symmetric polynomials, or geometric inequalities like $\text{Area}(A)\geq\text{Area}(B)$ when $A\supset B$, etc. and then you use very evil properties of the sign $\geq$, plus some algebraic massaging to the expressions you get.

The problem is that the last two steps, while very easy to do, can be very hard to reverse.

Using sharp bounds (e.g. $x^2\geq0$) as the building blocks helps to get a nicer looking problem, but you also have the freedom of not using sharp inequalities and this can make the problem arbitrarily difficult to solve. Suppose you start with $x^2\geq a$ with $a$ a negative number, but hard to determine that is negative.