2017 IMO Shortlist A1

Let $a_{1}, a_{2}, \ldots, a_{n}, k$, and $M$ be positive integers such that $$\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}}=k \quad \text { and } \quad a_{1} a_{2} \ldots a_{n}=M$$ If $M>1$, prove that the polynomial $$P(x)=M(x+1)^{k}-\left(x+a_{1}\right)\left(x+a_{2}\right) \cdots\left(x+a_{n}\right)$$ has no positive roots. (Trinidad and Tobago)

设 $a_{1}、a_{2}、\ldots、a_{n}、k$ 和 $M$ 为正整数，使得 $$\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}}=k \quad \text { 和 } \quad a_{1} a_{2} \ldots a_{n}=M$$如果 $M>1$，证明多项式 $$P(x)=M(x+1)^{k}-\left(x+a_{1}\right)\left(x+a_{2}\right) \cdots\left(x+a_{n}\right)$$ 无正根。 (特立尼达和多巴哥)