Question 9
Let $x_1,x_2,\ldots,x_{100}$ be positive real numbers. Let
$$P_i=x_i^{(\ln x_1)(\ln x_2)\cdots(\ln x_{i-1})(\ln x_{i+1})\cdots(\ln x_{100})}$$
for $i=1,2,\ldots,100$. Suppose that $\sum_{k=1}^{100}P_k=10000$. Find the value of $\left\lfloor\prod_{k=1}^{100}\ln x_k\right\rfloor$.
$$P_i=x_i^{(\ln x_1)(\ln x_2)\cdots(\ln x_{i-1})(\ln x_{i+1})\cdots(\ln x_{100})}$$
for $i=1,2,\ldots,100$. Suppose that $\sum_{k=1}^{100}P_k=10000$. Find the value of $\left\lfloor\prod_{k=1}^{100}\ln x_k\right\rfloor$.