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Complex roots and equations | AIME I, 1994 | Question 13

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1994 based on Complex roots and equations.

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1995 based on Complex roots and equations.

Complex roots and equations – AIME I, 1994


\(x^{10}+(13x-1)^{10}=0\) 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 complex conjugates are taken, find the values of \(\frac{1}{(r_1)(\overline{r_1})}+\frac{1}{(r_2)(\overline{r_2})}+\frac{1}{(r_3)(\overline{r_3})}+\frac{1}{(r_4)(\overline{r_4})}+\frac{1}{(r_5)(\overline{r_5})}\)

  • is 107
  • is 850
  • is 840
  • cannot be determined from the given information

Key Concepts


Integers

Complex Roots

Equation

Check the Answer


Answer: is 850.

AIME I, 1994, Question 13

Complex Numbers from A to Z by Titu Andreescue

Try with Hints


here equation gives \({13-\frac{1}{x}}^{10}=(-1)\)

\(\Rightarrow \omega^{10}=(-1)\) for \(\omega=13-\frac{1}{x}\)

where \(\omega=e^{i(2n\pi+\pi)(\frac{1}{10})}\) for n integer

\(\Rightarrow \frac{1}{x}=13- {\omega}\)

\(\Rightarrow \frac{1}{(x)(\overline{x})}=(13-\omega)(13-\overline{\omega})\)

=\(170-13(\omega+\overline{\omega})\)

adding over all terms \(\frac{1}{(r_1)(\overline{r_1})}+\frac{1}{(r_2)(\overline{r_2})}+\frac{1}{(r_3)(\overline{r_3})}+\frac{1}{(r_4)(\overline{r_4})}+\frac{1}{(r_5)(\overline{r_5})}\)

=5(170)

=850.

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