Q:

Prove that at least two of the inequalities are true.

Let a, b, and c be positive real numbers such that   $a + b + c\leq 4$ and
$ab+bc+ca\geq 4$.
Prove that at least two of the inequalities
$|a - b|\leq2$   , $|b - c| \leq 2$,   $ |c - a| \leq 2$
are true.

A:

You can deduce that
$$ (a-b)^2+(b-c)^2+(c-a)^2 \leq 8$$
If two of the inequalities you mention are false, then this is contradicted.

