Given positive integer \(n,k\) such that \(2 \le n < 2^k\). Prove that there exist a subset \(A\) of \(\{0,1,\cdots,n\}\) such that for any \(x \neq y \in A\), \({\binom{y}{x}}\) is even, and\[|A| \ge \frac{\binom{k}{\lfloor k/2 \rfloor}}{2^k} \cdot (n+1)\]