JMO 1995 Problem 5

Let $k,n$ be integers such that $1\leq k\leq n,$ a nd let $a_1,\ a_2,\ \cdots,\ a_k$ be numbers satisfying the following equations.

$$\begin{cases} a_1+\cdots\cdots +a_k=n \\ a_1^2+\cdots\cdots +a_k^2=n\\ \ \ \ \ \ \ \ \ \ \ \ \ \vdots \\ a_1^k+\cdots\cdots +a_k^k=n \end{cases}$$

Prove that $$(x+a_1)(x+a_2)\cdots(x+a_k)=x^k+\ _n C_1\ x^{k-1}+\ _nC_2\ x^{k-2}+\cdots+\ _nC_k.$$

where $_iC_j$ is a binomial coefficient which means $\frac{i\cdot (i-1)\cdots(i-j+1)}{j\cdot (j-1)\cdots 2\cdot 1}.$