Let S(x) denotes the sum of digits of x.If m and n are two positive integers such that $(S(m))^2 =n$ and $(S(n))^2 =m$ is true.Then find m,n. I made my start to this question such that $S(m)\equiv m\pmod 9$ so $n\equiv m^2\pmod 9$ and $m≡n^2(mod 9)$ so $n≡m^2≡n^4(mod 9)$ so $n^4-n≡0(mod 9)$ so e...

Given the polynomial $x^n - 3x^{n-1} + 2x + 1$ with following roots $x_1, x_2, ..., x_n$, find the sum $S = \sum_{k=1}^n\frac{x_k}{x_k-1}.$ I imagine it's some Vieta's involved, I tried splitting the sum as so: $S = \sum_{k=1}^n\left(\frac{x_k-1+1}{x_k-1}\right) = n - \sum_{k=1}^n\frac{1}{x_k-1}$...