MathNet 2025 — 50 problems — Part 01 Question 19

Suppose $a$, $b$, $c$ are real numbers such that $a + b + c = 1$ and $a^2 + b^2 + c^2 = 1$. Prove that $a^3 + b^3 + c^3 \ge \frac{5}{9}$.