$x_1 + x_2 + \dots + x_n = 1, 0 \leq x_i \leq 1$, and $(x_1, x_2, \dots, x_n)$ evenly distributes on its restricted space, obviously which is a polygon on $n - 1$ dimension plane.
Let random variable $Z = \max(x_1, x_2, \dots, x_n)$, what is the probability distribution function $F(m) = P(Z < m)$.
Obviously, $F\left(\frac{1}{n}\right) = 0, F(1) = 1$.
Dose this problem have a analytical solution? If not, Dose this problem have a recursive solution?
It is now a textbook fact that the joint distribution of your random variables $x_1,\dots,x_n$ is the same as that of $R_1,\dots,R_n$, where $R_i:=H_i/(H_1+\dots+H_n)$ and the $H_i$'s are iid standard exponential random variables. Moran, page 93 ascribes mentioning of this fact to Fisher, and a proof of it -- without a specific reference -- to Clifford.
Fisher also gave the distribution of $\max_i R_i$, which is, by what was just stated, the same as the distribution of $\max_i x_i$: $$P(\max_i x_i<u)=P(\max_i R_i<u) =\sum_{j=0}^n(-1)^j\binom nj(1-ju)_+^{n-1} $$ for $u\in[0,1]$, where $v_+:=\max(0,v)$.
