MathNet 2005 — Part 03 Question 4

Let $a$, $b$, $c$, $d$ be positive real numbers such that $a + b + c + d = 1$. Prove that
$$
6(a^3 + b^3 + c^3 + d^3) \ge (a^2 + b^2 + c^2 + d^2) + \frac{1}{8}.
$$