Let $p>2$ be an odd prime number. Integers $a_1,a_2,\ldots, a_{p+1}$ in the interval $[0,p]$ have the following property: for every permutation $\pi$ of the set $\{1,2,\ldots,p+1\}$ the number $$\sum_{k=1}^{p+1}ka_{\pi(k)}$$ is not divisible by $p$. Prove that $a_1=a_2=\ldots=a_{p+1}$.