BALTIC-WAY 2004 Problem 1

Given a sequence $a_{1}, a_{2}, a_{3}, \ldots$ of non-negative real numbers satisfying the conditions

(1) $a_{n}+a_{2 n} \geq 3 n$

(2) $a_{n+1}+n \leq 2 \sqrt{a_{n} \cdot(n+1)}$

for all indices $n=1,2 \ldots$

(a) Prove that the inequality $a_{n} \geq n$ holds for every $n \in \mathbb{N}$.

(b) Give an example of such a sequence.