MathNet 2007-part-01 — 187 problems (EN–VI) Question 122

For nonnegative real numbers $a$, $b$, $c$ with $a + b + c = 1$, prove that
$$
\sqrt{a + \frac{(b-c)^2}{4}} + \sqrt{b} + \sqrt{c} \le \sqrt{3}.
$$