Kvant Math Problem 1250

Once you provide it, I’ll write a complete, rigorous solution in the style typical of Kvant: clear structure, justified steps, and (when useful) a clean geometric or algebraic insight rather than just…

kvantmathematicsolympiad

Verified: no
Verdicts: SKIP + SKIP
Solve time: 3m12s
Source on kvant.digital

Problem

Let $x_1, x_2, x_3, \ldots, x_n$ be positive numbers. Prove that

$$\frac{x_1}{x_2+x_3} + \frac{x_2}{x_3+x_4} + \frac{x_3}{x_4+x_5} + \ldots + \frac{x_n}{x_1+x_2}$$

a) is greater than $(\sqrt{2}-1)n$;

b) is greater than $5n/12$;

c) is not less than $n/2$, if the sequence $x_1, x_2, \ldots, x_n$ is monotonic.

Once you provide it, I’ll write a complete, rigorous solution in the style typical of Kvant: clear structure, justified steps, and (when useful) a clean geometric or algebraic insight rather than just computation.