Question 41
Let $n$, $k$, $M$ and $a_1, a_2, \dots, a_n$ be positive integers such that
$$
\frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n} = k \quad \text{and} \quad a_1 a_2 \dots a_n = M.
$$
If $M \gt 1$, prove that there does not exist a positive real number $x$ such that
$$
M(x+1)^k = (x+a_1)(x+a_2)\dots(x+a_n).
$$
$$
\frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n} = k \quad \text{and} \quad a_1 a_2 \dots a_n = M.
$$
If $M \gt 1$, prove that there does not exist a positive real number $x$ such that
$$
M(x+1)^k = (x+a_1)(x+a_2)\dots(x+a_n).
$$