Complexity Is Awesome!
Algebra
Level
5
The equation \(x^{10}+(13x-1)^{10}=0\) has \(10\) complex roots: \(r_1, \overline{r_1}, r_2, \overline{r_2}, r_3, \overline{r_3}, r_4, \overline{r_4}, r_5, \overline{r_5},\) where the bar denotes complex conjugation. Evaluate the expression below.
\[\dfrac 1{r_1\overline{r_1}}+\dfrac 1{r_2\overline{r_2}}+\dfrac 1{r_3\overline{r_3}}+\dfrac 1{r_4\overline{r_4}}+\dfrac 1{r_5\overline{r_5}} \]
Image Credit: Wikimedia Decagon by László Németh
by
Parth Lohomi