<s>[INST] Prove the inequalitiy 
If $a, b, c$ are positive reals such that:
  $\frac a {b + c + 1} + \frac b {a + c + 1} + \frac c {b + a + 1} \le
 1$ then:
$\frac 1 {b + c + 1} + \frac 1 {a + c + 1} + \frac 1 {b + a + 1} \ge
 1$


I tried using inequality of arithmetic and harmonic means, to no avail.
Any help is appreciated.
 [/INST] We'll write the condition in the following form:
$$a+b+c+1\geq a^3+b^3+c^3+abc+\sum\limits_{cyc}(a^2-ab)$$
and since $\sum\limits_{cyc}(a^2-ab)\geq0$ we obtain
$$a+b+c+1\geq a^3+b^3+c^3+abc$$
In another hand, we need to prove that 
$$2(a+b+c+1)\geq(a+b)(a+c)(b+c)$$
Thus, it remains to prove that
$$2(a^3+b^3+c^3+abc)\geq(a+b)(a+c)(b+c)$$
which is $\sum\limits_{cyc}(a+b)(a-b)^2\geq0$. Done!
</s>