All-Soviet-Union Mathematical Olympiad

Solution:

We have
$$\sum \frac{x_i^2}{x_i + x_{i+1}} - \sum \frac{x_{i+1}^2}{x_i + x_{i+1}} = \sum (x_i - x_{i+1}) = 0.$$
Also,
$$\frac{x_i^2 + x_{i+1}^2}{x_i + x_{i+1}} \geq \frac{1}{2} (x_i + x_{i+1})$$
by the QM-AM inequality. Summing over all $i$,
$$\sum \frac{x_i^2}{x_i + x_{i+1}} + \sum \frac{x_{i+1}^2}{x_i + x_{i+1}} \geq \frac{1}{2} \sum (x_i + x_{i+1}) = 1.$$
Since the two sums are equal, each is at least $\frac{1}{2}$.
Therefore,
$$\sum \frac{x_i^2}{x_i + x_{i+1}} \geq \frac{1}{2}.$$