ISL 1979 Problem 40

A polynomial $P(x)$ has degree at most $2k$ , where $k = 0, 1,2,\cdots$ . Given that for an integer $i$ , the inequality $-k \le i \le k$ implies $|P(i)| \le 1$ , prove that for all real numbers $x$ , with $-k \le x \le k$ , the following inequality holds: $$|P(x)| < (2k + 1)\dbinom{2k}{k}$$