Explanation:
Okay, so I need to solve this equation: $x^{10} + (13x - 1)^{10} = 0$. The problem says it has 10 complex roots, which are paired as conjugates: $r_1, \overline{r}_1, r_2, \overline{r}_2,$ and so on up to $r_5, \overline{r}_5$. Then I need to find the sum $\frac{1}{r_1\overline{r}_1} + \frac{1}{r_2\overline{r}_2} + \cdots + \frac{1}{r_5\overline{r}_5}$.

First, let me recall that for any complex number $z$, the product $z \overline{z}$ is equal to $|z|^2$, the square of its modulus. So each term in the sum is $1/|r_i|^2$. Therefore, the problem is asking for the sum of the reciprocals of the squared moduli of the roots.

But how do I find that? Maybe I can relate this to the coefficients of the polynomial? Since the polynomial is given, perhaps I can use Vieta's formulas or some other properties of polynomials?

Let me first try to understand the equation better. The equation is $x^{10} + (13x - 1)^{10} = 0$. Let me rewrite this as:

$$
x^{10} = - (13x - 1)^{10}
$$

Taking both sides to the power of 1/10, but since we are dealing with complex numbers, there are multiple roots. However, maybe I can write this as:

$$
\left( \frac{x}{13x - 1} \right)^{10} = -1
$$

So, $\frac{x}{13x - 1}$ is a 10th root of -1. Let me denote $y = \frac{x}{13x - 1}$. Then $y^{10} = -1$. Therefore, $y$ is a complex number such that $y^{10} = -1$. The solutions to this equation are the 10th roots of -1, which can be written as $y = e^{i(\pi + 2\pi k)/10}$ for $k = 0, 1, ..., 9$.

Alternatively, since $-1 = e^{i\pi}$, the 10th roots of -1 are $e^{i(\pi + 2\pi k)/10}$ for $k = 0$ to 9. So each $y$ is a complex number on the unit circle at angles $\pi/10, 3\pi/10, ..., 19\pi/10$.

But maybe there's a better way to approach this. Let me try to express $x$ in terms of $y$. From $y = \frac{x}{13x - 1}$, solving for $x$:

Multiply both sides by $13x - 1$:

$$
y(13x - 1) = x \\
13y x - y = x \\
13y x - x = y \\
x(13y - 1) = y \\
x = \frac{y}{13y - 1}
$$

So each root $x$ of the original equation can be written as $x = \frac{y}{13y - 1}$, where $y^{10} = -1$. Therefore, the roots $x$ are given by this expression for each 10th root of -1.

But since the original equation is degree 10, there are 10 roots, which matches with the 10 values of $y$. However, the problem mentions that the roots come in conjugate pairs. So perhaps some of these $y$ values are conjugates of each other? Let me check.

If $y$ is a root of $y^{10} = -1$, then its complex conjugate $\overline{y}$ is also a root because if $y^{10} = -1$, then $\overline{y}^{10} = \overline{(-1)} = -1$. Therefore, the roots $y$ come in conjugate pairs. Therefore, when we take $x = \frac{y}{13y - 1}$, then for each pair $y$ and $\overline{y}$, we get two roots $x$ and $\overline{x}$, since substituting $\overline{y}$ into the expression would give $\overline{x}$. Therefore, the roots $x$ also come in conjugate pairs, as stated in the problem.

Therefore, each pair $r_i$ and $\overline{r}_i$ corresponds to a pair of $y$ values $y$ and $\overline{y}$. Therefore, perhaps I can compute $1/(r_i \overline{r}_i)$ for each pair and sum them up.

But how do I compute $1/(r_i \overline{r}_i)$? Since $r_i \overline{r}_i = |r_i|^2$, so $1/|r_i|^2$. Alternatively, since $x = \frac{y}{13y - 1}$, then $|x|^2 = \frac{|y|^2}{|13y - 1|^2}$. Therefore, $1/|x|^2 = \frac{|13y - 1|^2}{|y|^2}$. Therefore, for each root $x$ corresponding to a $y$, $1/|x|^2 = \frac{|13y - 1|^2}{|y|^2}$.

But since $y$ is on the unit circle (because $y^{10} = -1$, so $|y| = 1$), then $|y|^2 = 1$. Therefore, $1/|x|^2 = |13y - 1|^2$. Therefore, the sum we need is the sum over all pairs $r_i, \overline{r}_i$ of $|13y - 1|^2$, but since each pair corresponds to a $y$ and $\overline{y}$, and since $|13y - 1|^2 = |13\overline{y} - 1|^2$ (because modulus squared is invariant under conjugation), then each pair contributes twice the value of $|13y - 1|^2$? Wait, no. Wait, each pair $y$ and $\overline{y}$ corresponds to two roots $x$ and $\overline{x}$, so each pair contributes two terms in the sum: $1/|x|^2$ and $1/|\overline{x}|^2$, but since $1/|\overline{x}|^2 = 1/|x|^2$, so each pair contributes twice $1/|x|^2$. However, since we have 5 pairs, the total sum is 2 times the sum over each pair's $1/|x|^2$. Wait, but actually, the original sum is over 5 terms, each being $1/(r_i \overline{r}_i)$. Since each pair $r_i$ and $\overline{r}_i$ contributes one term $1/(r_i \overline{r}_i)$, which is equal to $1/|r_i|^2$. Therefore, since each pair corresponds to two roots $x$ and $\overline{x}$, but the term $1/|x|^2$ is the same for both, so each pair contributes one term. Therefore, the total sum is the sum over all 5 pairs of $1/|x|^2$, which is equal to the sum over all 10 roots of $1/|x|^2$ divided by 2? Wait, no. Wait, each pair contributes one term, so the total sum is the sum over each pair's $1/|x|^2$. But since each pair corresponds to two roots, but the term is the same for both, so the total sum is the sum over all roots of $1/|x|^2$ divided by 2? Wait, no. Let me think again.

Suppose there are 10 roots, each paired with their conjugate. So for each pair $r_i$ and $\overline{r}_i$, we have two roots. The sum given in the problem is over 5 terms, each being $1/(r_i \overline{r}_i)$. Since $1/(r_i \overline{r}_i) = 1/|r_i|^2$, and since $1/|r_i|^2 = 1/| \overline{r}_i |^2$, so each pair contributes one term. Therefore, the total sum is the sum over all 5 pairs of $1/|r_i|^2$. Therefore, if I can compute $1/|x|^2$ for each root $x$, and then sum them up, but since each pair contributes the same value twice? Wait, no. Wait, if I have two roots $x$ and $\overline{x}$, then $1/|x|^2$ and $1/|\overline{x}|^2$ are the same, so each pair contributes twice the same value? Wait, no. Wait, the problem says the sum is over 5 terms, each being $1/(r_i \overline{r}_i)$. So each pair contributes one term. Therefore, if I can compute $1/|x|^2$ for each root, and then sum them up, but since each pair is counted once, then the total sum is the sum over all roots of $1/|x|^2$ divided by 2? Wait, no. Let me think with an example. Suppose there are two roots, $r$ and $\overline{r}$. Then the sum would be $1/(r \overline{r}) + 1/(\overline{r} r) = 2/(r \overline{r})$. But in the problem, the sum is over 5 terms, each being $1/(r_i \overline{r}_i)$. Therefore, if there are 5 pairs, then the total sum is 5 terms, each being $1/(r_i \overline{r}_i)$. Therefore, if I have 10 roots, each pair contributes one term. Therefore, the total sum is equal to the sum over all roots of $1/|x|^2$ divided by 2? Wait, no. If each pair contributes one term, then the total sum is equal to the sum over all roots of $1/|x|^2$ divided by 2? Wait, no. Let me think again. Suppose there are two roots, $r$ and $\overline{r}$. Then the sum is $1/(r \overline{r})$, which is $1/|r|^2$. However, if I compute $1/|r|^2 + 1/|\overline{r}|^2$, that would be $2/|r|^2$. Therefore, if the problem is asking for the sum over the pairs, which is 1 term per pair, then it's $1/|r|^2$, but if I compute the sum over all roots, it would be $2/|r|^2$. Therefore, in our case, the problem is asking for the sum over 5 terms, each being $1/|r_i|^2$, so that would be half of the sum over all roots of $1/|x|^2$. Therefore, if I can compute the sum over all roots of $1/|x|^2$, then divide by 2, I would get the desired value.

But how do I compute the sum over all roots of $1/|x|^2$? Alternatively, since $1/|x|^2 = |1/x|^2$, so the sum is the sum over all roots of $|1/x|^2$. Therefore, if I can compute the sum of $|1/x|^2$ over all roots, then divide by 2, I get the answer.

Alternatively, since $x$ satisfies the equation $x^{10} + (13x - 1)^{10} = 0$, which can be rewritten as $(13x - 1)^{10} = -x^{10}$. Taking modulus on both sides, we get $|13x - 1|^{10} = |x|^{10}$. Therefore, $|13x - 1| = |x|$. Therefore, $|13x - 1|^2 = |x|^2$. Therefore, $|13x - 1|^2 = |x|^2$. Wait, but earlier I had that $1/|x|^2 = |13y - 1|^2$, where $y = x/(13x - 1)$. But if $|13x - 1|^2 = |x|^2$, then $|13x - 1|^2 / |x|^2 = 1$, so $1/|x|^2 = |13x - 1|^2 / |x|^2 * |x|^2 / |x|^2$... Wait, maybe I need to check again.

Wait, earlier I had that $x = y/(13y - 1)$, and since $y^{10} = -1$, and $|y| = 1$. Then $|x|^2 = |y|^2 / |13y - 1|^2 = 1 / |13y - 1|^2$. Therefore, $1/|x|^2 = |13y - 1|^2$. Therefore, the sum over all roots of $1/|x|^2$ is equal to the sum over all $y$ (which are the 10th roots of -1) of $|13y - 1|^2$. Therefore, since each root $x$ corresponds to a unique $y$, and there are 10 roots, the sum over all roots of $1/|x|^2$ is equal to the sum over all $y$ of $|13y - 1|^2$. Therefore, the desired sum (which is half of that) is equal to half of the sum over all $y$ of $|13y - 1|^2$.

Therefore, if I can compute the sum over all 10th roots of -1 of $|13y - 1|^2$, then divide by 2, I get the answer.

So let me compute that sum. Let me denote $S = \sum_{y^{10} = -1} |13y - 1|^2$. Then the answer is $S / 2$.

Now, how do I compute $S$? Let me recall that for any complex number $z$, $|z|^2 = z \overline{z}$. Therefore, $|13y - 1|^2 = (13y - 1)(13\overline{y} - 1)$. Since $y^{10} = -1$, then $\overline{y} = y^{-10} \cdot (-1)^{-1}$. Wait, maybe not. Wait, since $y^{10} = -1$, then taking modulus squared on both sides, $|y|^{10} = |-1| = 1$, so $|y| = 1$. Therefore, $\overline{y} = 1/y$. Therefore, $|13y - 1|^2 = (13y - 1)(13\overline{y} - 1) = (13y - 1)(13/y - 1)$. Let me compute that:

$$
(13y - 1)\left( \frac{13}{y} - 1 \right) = 13y \cdot \frac{13}{y} - 13y \cdot 1 - 1 \cdot \frac{13}{y} + 1 \cdot 1 = 169 - 13y - \frac{13}{y} + 1 = 170 - 13\left(y + \frac{1}{y}\right)
$$

Therefore, $|13y - 1|^2 = 170 - 13\left(y + \frac{1}{y}\right)$. Therefore, the sum $S$ becomes:

$$
S = \sum_{y^{10} = -1} \left[170 - 13\left(y + \frac{1}{y}\right)\right] = 10 \times 170 - 13 \sum_{y^{10} = -1} \left(y + \frac{1}{y}\right)
$$

So, $S = 1700 - 13 \times \left[ \sum_{y^{10} = -1} y + \sum_{y^{10} = -1} \frac{1}{y} \right]$.

But since $y$ are the 10th roots of -1, which are the same as the 20th roots of 1 that are not 10th roots of 1. Wait, maybe not. Let me think. The equation $y^{10} = -1$ can be rewritten as $y^{20} = 1$, so the solutions are the 20th roots of unity, but only those that satisfy $y^{10} = -1$, which are the primitive 20th roots of unity and the ones with order dividing 20 but not dividing 10. Wait, more precisely, the solutions to $y^{10} = -1$ are the 20th roots of unity that are not 10th roots of unity. Because if $y^{10} = -1$, then $y^{20} = 1$, so they are 20th roots of unity. However, the 10th roots of unity satisfy $y^{10} = 1$, so the solutions to $y^{10} = -1$ are the 20th roots of unity that are not 10th roots of unity. Therefore, there are 10 such roots.

Alternatively, since $y^{10} = -1$, we can write $y = e^{i\theta}$, where $10\theta = \pi + 2\pi k$, so $\theta = \pi/10 + \pi k/5$, for $k = 0, 1, ..., 9$. Therefore, the roots are $e^{i(\pi/10 + \pi k/5)}$ for $k = 0$ to 9. Let me check for k=0: $\pi/10$, k=1: $\pi/10 + \pi/5 = 3\pi/10$, k=2: $\pi/10 + 2\pi/5 = 5\pi/10 = \pi/2$, k=3: $\pi/10 + 3\pi/5 = 7\pi/10$, k=4: $\pi/10 + 4\pi/5 = 9\pi/10$, k=5: $\pi/10 + \pi = 11\pi/10$, k=6: $\pi/10 + 6\pi/5 = 13\pi/10$, k=7: $\pi/10 + 7\pi/5 = 15\pi/10 = 3\pi/2$, k=8: $\pi/10 + 8\pi/5 = 17\pi/10$, k=9: $\pi/10 + 9\pi/5 = 19\pi/10$. So these are the angles for the roots.

But maybe I don't need to go into angles. Let me think about the sum $\sum_{y^{10} = -1} y$. Since the roots of $y^{10} + 1 = 0$ are the roots of $y^{10} = -1$, which is a polynomial of degree 10. Therefore, the sum of the roots is given by Vieta's formula. The polynomial is $y^{10} + 1 = 0$, which can be written as $y^{10} + 0y^9 + 0y^8 + \cdots + 0y + 1 = 0$. Therefore, the sum of the roots is $-0/1 = 0$. Therefore, $\sum_{y^{10} = -1} y = 0$. Similarly, $\sum_{y^{10} = -1} \frac{1}{y}$. Since $y$ are roots of $y^{10} + 1 = 0$, then $1/y$ are roots of $1 + y^{10} = 0$, which is the same equation. Therefore, the sum $\sum \frac{1}{y}$ is also equal to the sum of the roots of $y^{10} + 1 = 0$, which is 0. Therefore, both sums are zero. Therefore, $S = 1700 - 13 \times (0 + 0) = 1700$. Therefore, the sum over all roots of $1/|x|^2$ is 1700, so the desired sum is 1700 / 2 = 850. Wait, but hold on. Wait, the problem says the answer is the sum over 5 terms, each being $1/(r_i \overline{r}_i)$. But according to this, the sum over all roots of $1/|x|^2$ is 1700, so the sum over the pairs would be 1700 / 2 = 850. Therefore, the answer is 850? But let me check again.

Wait, earlier I said that the sum over all roots of $1/|x|^2$ is equal to S = 1700, and since each pair contributes one term, then the answer is 1700 / 2 = 850. But let me check again the logic.

Each root x corresponds to a y, and for each pair of roots x and \overline{x}, they correspond to y and \overline{y}. Therefore, the sum over all roots of 1/|x|^2 is equal to the sum over all y of |13y - 1|^2, which is S = 1700. However, since each pair (x, \overline{x}) contributes two roots, but the sum over all roots includes both. However, the problem asks for the sum over 5 terms, each being 1/(r_i \overline{r}_i). Since each pair contributes one term, which is 1/|r_i|^2, and since there are 5 pairs, the total sum is 5 terms. However, the sum over all roots of 1/|x|^2 is 2 times the sum over the pairs. Because each pair contributes two roots, each with the same 1/|x|^2. Therefore, if the total sum over all roots is 1700, then the sum over the pairs is 1700 / 2 = 850. Therefore, the answer is 850.

But let me check with a simpler case. Suppose I have two roots, r and \overline{r}. Then the sum over all roots of 1/|x|^2 is 1/|r|^2 + 1/|\overline{r}|^2 = 2/|r|^2. The sum over the pairs is just 1/|r|^2. Therefore, yes, the sum over the pairs is half of the total sum over all roots. Therefore, in our case, since there are 10 roots, the sum over the pairs is 1700 / 2 = 850. Therefore, the answer is 850.

But let me check if my calculation of S is correct. Let me recap:

We had S = sum_{y^{10} = -1} |13y - 1|^2 = 1700 - 13*(sum y + sum 1/y). Then since sum y = 0 and sum 1/y = 0, so S = 1700.

But why is sum y = 0? Because the polynomial is y^{10} + 1 = 0, which is y^{10} + 0y^9 + ... + 0y + 1 = 0. Therefore, the sum of roots is -coefficient of y^9 / coefficient of y^{10} = -0/1 = 0. Similarly, sum 1/y is the same as sum of roots of the polynomial obtained by reversing coefficients. The polynomial is y^{10} + 1 = 0. Reversing coefficients gives 1 + y^{10} = 0, which is the same equation. Therefore, the sum of 1/y is also 0. Therefore, yes, S = 1700.

Therefore, the answer is 850. However, let me check with another approach to make sure.

Alternative approach: Let me consider the original equation $x^{10} + (13x - 1)^{10} = 0$. Let me divide both sides by $x^{10}$ (assuming x ≠ 0, which is valid since if x=0, then 0 + (-1)^10 = 1 ≠ 0, so x=0 is not a root). Therefore, we get:

$$
1 + \left(13 - \frac{1}{x}\right)^{10} = 0
$$

Therefore,

$$
\left(13 - \frac{1}{x}\right)^{10} = -1
$$

Let me denote $z = 13 - \frac{1}{x}$. Then $z^{10} = -1$, so z is a 10th root of -1, same as before. Therefore, $z = e^{i(\pi + 2\pi k)/10}$ for k=0,...,9. Then solving for x:

$$
z = 13 - \frac{1}{x} \implies \frac{1}{x} = 13 - z \implies x = \frac{1}{13 - z}
$$

Therefore, each root x is given by $x = 1/(13 - z)$, where z^{10} = -1. Therefore, similar to before, but with a different expression.

Now, let me compute $1/|x|^2 = |13 - z|^2$. Therefore, the sum over all roots of $1/|x|^2$ is equal to sum_{z^{10}=-1} |13 - z|^2. Which is the same as before, since earlier we had |13y - 1|^2, but here it's |13 - z|^2. However, since z is a root of z^{10} = -1, which is same as y^{10} = -1. Therefore, the sum is same as before. Therefore, same result. Therefore, S = 1700. Therefore, the answer is 850.

But let me check with specific values. Let me take a simple case. Suppose I take a root z such that z^{10} = -1. Let me pick z = e^{i\pi/10}. Then compute |13 - z|^2. Similarly for other roots. But since the sum over all roots is 1700, which is 10*170, and since each term is 170 - 13*(z + 1/z), and sum over z + 1/z is zero. Therefore, the total sum is 10*170 = 1700. Therefore, seems correct.

Alternatively, maybe I can compute the sum using another method. Let me recall that for any complex number z, |a - z|^2 = |a|^2 + |z|^2 - 2 Re(a \overline{z}). Therefore, |13 - z|^2 = 13^2 + |z|^2 - 2 Re(13 \overline{z}). Since |z| = 1 (because z^{10} = -1 implies |z|^{10} = 1, so |z| = 1). Therefore, |13 - z|^2 = 169 + 1 - 2 Re(13 \overline{z}) = 170 - 26 Re(\overline{z}) = 170 - 26 Re(z). Because Re(\overline{z}) = Re(z). Therefore, sum_{z^{10}=-1} |13 - z|^2 = sum_{z} [170 - 26 Re(z)] = 10*170 - 26 sum Re(z).

But sum Re(z) over all z^{10} = -1 is equal to Re(sum z) = Re(0) = 0. Therefore, the total sum is 1700 - 0 = 1700. Therefore, same result. Therefore, this confirms that S = 1700. Therefore, the answer is 850.

But let me check once more. Suppose I take a specific root. Let me take z = e^{i\pi/10}. Then compute |13 - z|^2. But even if I compute it, the sum over all roots would still be 1700. Therefore, seems correct.

Therefore, the answer is 850. However, let me check if there is a mistake in the logic.

Wait, the problem says that the roots are r1, \overline{r1}, ..., r5, \overline{r5}. Therefore, there are 10 roots. The sum we need is sum_{i=1}^5 1/(r_i \overline{r_i}) = sum_{i=1}^5 1/|r_i|^2. But since each pair contributes two roots, but the term is same for both, so the total sum is sum_{i=1}^5 1/|r_i|^2. However, the sum over all roots of 1/|x|^2 is 2 * sum_{i=1}^5 1/|r_i|^2. Therefore, the answer is S / 2 = 1700 / 2 = 850. Therefore, yes.

Alternatively, maybe I can think in terms of polynomials. Let me consider the original equation: x^{10} + (13x - 1)^{10} = 0. Let me denote this polynomial as P(x) = x^{10} + (13x - 1)^{10}. Let me consider the polynomial Q(x) = P(x). The roots of Q(x) are the roots we are considering.

We need to compute sum_{i=1}^5 1/(r_i \overline{r_i}) = sum_{i=1}^5 1/|r_i|^2.

But for a polynomial with roots r_i, the sum of 1/|r_i|^2 can be related to the coefficients? Let me think.

Note that 1/|r_i|^2 = |1/r_i|^2. Therefore, if I consider the polynomial whose roots are 1/r_i, then the sum of |1/r_i|^2 is the sum of the squares of the moduli of the roots of that polynomial.

But how to find that polynomial? Let me try. Let me denote y = 1/x. Then x = 1/y. Substitute into P(x) = 0:

(1/y)^{10} + (13*(1/y) - 1)^{10} = 0

Multiply both sides by y^{10}:

1 + (13 - y)^{10} = 0

Therefore, the polynomial equation for y is (13 - y)^{10} + 1 = 0, which is same as (y - 13)^{10} + 1 = 0. Therefore, the roots y_i = 1/r_i satisfy (y_i - 13)^{10} = -1. Therefore, similar to before. Therefore, the roots y_i are 13 - z, where z^{10} = -1. Therefore, same as before. Therefore, the roots y_i are 13 - z, where z^{10} = -1. Therefore, the sum of |y_i|^2 is sum |13 - z|^2, which is S = 1700. Therefore, the sum of |y_i|^2 is 1700. However, the sum we need is sum_{i=1}^5 1/|r_i|^2 = sum_{i=1}^5 |y_i|^2. Wait, because 1/|r_i|^2 = |y_i|^2. Therefore, the sum we need is sum |y_i|^2. However, the sum over all roots y_i of |y_i|^2 is 1700. But since there are 10 roots y_i, and they come in conjugate pairs, then the sum over all roots of |y_i|^2 is 2 * sum_{i=1}^5 |y_i|^2. Therefore, the sum we need is 1700 / 2 = 850. Therefore, same result.

Therefore, this confirms again that the answer is 850.

But let me check with Vieta's formula. Suppose I want to compute sum |y_i|^2. For complex numbers, |y_i|^2 = y_i \overline{y_i}. Therefore, sum |y_i|^2 = sum y_i \overline{y_i}. However, since the roots y_i are 13 - z, where z^{10} = -1. Therefore, the roots y_i are 13 - z, and their conjugates are 13 - \overline{z}. Therefore, for each pair y and \overline{y}, we have y = 13 - z and \overline{y} = 13 - \overline{z}. Therefore, the product y \overline{y} = |y|^2 = |13 - z|^2. Therefore, the sum over all roots of |y|^2 is sum |13 - z|^2 = 1700. Therefore, same as before.

Alternatively, if I consider the polynomial whose roots are y_i = 13 - z, where z^{10} = -1. Let me find the polynomial for y. Let me denote z^{10} = -1. Then, z^{10} + 1 = 0. Let me express z in terms of y: z = 13 - y. Therefore, substituting into z^{10} + 1 = 0:

(13 - y)^{10} + 1 = 0. Therefore, the polynomial is (y - 13)^{10} + 1 = 0. Therefore, the roots are y_i = 13 - z, as before.

Now, to compute sum |y_i|^2, which is sum y_i \overline{y_i}. However, since the polynomial has real coefficients? Wait, the polynomial (y - 13)^{10} + 1 = 0 has real coefficients. Therefore, the roots come in conjugate pairs. Therefore, the sum of y_i \overline{y_i} is equal to sum |y_i|^2.

But how can I compute this sum? For a polynomial with real coefficients, the sum of |y_i|^2 can be related to the coefficients. Let me recall that for a polynomial with roots y_i, sum |y_i|^2 = sum y_i \overline{y_i} = sum y_i \overline{y_i}. However, since the polynomial has real coefficients, the non-real roots come in conjugate pairs. Therefore, for each pair y and \overline{y}, their contribution to the sum is y \overline{y} + \overline{y} y = 2 |y|^2. Wait, no. Wait, if you have a pair y and \overline{y}, then the sum over their |y|^2 is |y|^2 + |\overline{y}|^2 = 2 |y|^2. However, if you have a real root, then |y|^2 = y^2. Therefore, the total sum is sum_{i=1}^{10} |y_i|^2.

But how to compute this sum? Let me recall that for any complex number, |y|^2 = y \overline{y}. Therefore, sum |y_i|^2 = sum y_i \overline{y_i}. However, since the polynomial has real coefficients, the sum over y_i \overline{y_i} can be related to the coefficients.

Alternatively, note that sum |y_i|^2 = sum y_i \overline{y_i} = sum y_i \overline{y_i} = sum y_i \overline{y_i}. But since the polynomial is (y - 13)^{10} + 1 = 0, which is a polynomial with real coefficients. Let me denote the polynomial as P(y) = (y - 13)^{10} + 1. Let me expand this polynomial. However, expanding it might be complicated, but maybe I can use Vieta's formula.

Alternatively, note that sum |y_i|^2 = sum y_i \overline{y_i} = sum y_i \overline{y_i} = sum y_i \overline{y_i}. However, since the coefficients are real, the sum over y_i \overline{y_i} is equal to the sum over y_i \overline{y_i} = sum y_i \overline{y_i} = sum y_i \overline{y_i}.

Alternatively, consider that for any polynomial with real coefficients, the sum of the squares of the moduli of the roots can be calculated using the formula:

sum |y_i|^2 = (sum y_i^2) + (sum \overline{y_i}^2) - 2 Re(sum y_i \overline{y_i})? Wait, no. Wait, maybe not. Let me think.

Alternatively, note that for any complex number, |y|^2 = y \overline{y}. Therefore, sum |y_i|^2 = sum y_i \overline{y_i}. However, since the polynomial has real coefficients, the sum over y_i \overline{y_i} is equal to the sum over y_i \overline{y_i} = sum y_i \overline{y_i}. But how to compute this?

Alternatively, consider that if we have a polynomial with roots y_i, then the sum of y_i \overline{y_i} can be related to the coefficients. However, since the polynomial has real coefficients, the sum over y_i \overline{y_i} is equal to the sum over y_i \overline{y_i} = sum y_i \overline{y_i}. But this seems circular.

Alternatively, consider that if we have a polynomial with real coefficients, then the sum over y_i \overline{y_i} is equal to the sum over y_i \overline{y_i} = sum y_i \overline{y_i} = sum y_i \overline{y_i}.

Alternatively, think of the polynomial whose roots are y_i and \overline{y_i}. But since the polynomial already has real coefficients, the roots come in conjugate pairs. Therefore, the sum over y_i \overline{y_i} is equal to the sum over |y_i|^2.

But perhaps I can use the fact that for a polynomial P(y) = \prod_{i=1}^{n} (y - y_i), then the sum of y_i \overline{y_i} can be related to the coefficients. However, this seems complicated.

Alternatively, since we already have that sum |y_i|^2 = 1700, from previous calculations, and that this is equal to the sum over all roots of |13 - z|^2, which we calculated as 1700, then we can trust that result. Therefore, the answer is 850.

But to be thorough, let me check with a small example. Let me take a simpler equation where I can compute manually. For example, suppose the equation is x^2 + (13x - 1)^2 = 0. Then, solving this would give x^2 + (13x - 1)^2 = 0 => x^2 + 169x^2 - 26x + 1 = 0 => 170x^2 - 26x + 1 = 0. The roots are [26 ± sqrt(676 - 680)] / 340. Wait, discriminant is 676 - 680 = -4. Therefore, roots are [26 ± 2i]/340 = [13 ± i]/170. Therefore, the roots are (13 + i)/170 and (13 - i)/170. Therefore, their moduli squared are [(13)^2 + 1^2]/(170^2) = (169 + 1)/28900 = 170/28900 = 1/170. Therefore, the sum of 1/|r_i|^2 is 1/(1/170) + 1/(1/170) = 170 + 170 = 340. However, according to our formula, S would be sum |13y - 1|^2 where y^2 = -1. The roots y are i and -i. Therefore, for y = i: |13i - 1|^2 = | -1 + 13i |^2 = 1 + 169 = 170. Similarly for y = -i: | -1 -13i |^2 = 1 + 169 = 170. Therefore, S = 170 + 170 = 340. Then the answer would be S / 2 = 170. However, in reality, the sum of 1/|r_i|^2 is 340. Wait, but according to our previous logic, the answer should be S / 2 = 340 / 2 = 170. But in reality, the sum of 1/|r_i|^2 is 340. Therefore, there is a contradiction here.

Wait, this suggests that my previous logic is wrong. What's happening here?

Wait, in this case, the original equation is x^2 + (13x - 1)^2 = 0. The roots are r and \overline{r}, so the sum we need is 1/(r \overline{r}) + 1/(\overline{r} r) = 2/(r \overline{r}). But according to the calculation, r \overline{r} = |r|^2 = 1/170, so the sum is 2/(1/170) = 340. However, according to the previous logic, S = sum |13y - 1|^2 = 340, and the answer is S / 2 = 170. But in reality, the answer is 340. Therefore, there is a mistake in my previous reasoning.

This suggests that my earlier conclusion that the answer is S / 2 is wrong. Therefore, I need to re-examine my logic.

Let me go back. In this simple case, the original equation is x^2 + (13x - 1)^2 = 0. The roots are x = [13 ± i]/170. Therefore, each root x corresponds to a y such that y = x/(13x - 1). Let me check.

For x = [13 + i]/170, compute y = x / (13x - 1). Let me compute 13x - 1: 13*(13 + i)/170 - 1 = (169 + 13i)/170 - 1 = (169 + 13i - 170)/170 = (-1 + 13i)/170. Therefore, y = [ (13 + i)/170 ] / [ (-1 + 13i)/170 ] = (13 + i)/(-1 + 13i). Multiply numerator and denominator by (-1 - 13i):

Numerator: (13 + i)(-1 - 13i) = -13 - 169i - i -13i^2 = -13 - 170i + 13 = -170i

Denominator: (-1)^2 - (13i)^2 = 1 - (-169) = 1 + 169 = 170

Therefore, y = (-170i)/170 = -i. Similarly, for the other root x = [13 - i]/170, y = i. Therefore, the roots y are i and -i, which satisfy y^2 = -1. Therefore, in this case, S = sum |13y - 1|^2 = |13i - 1|^2 + |13(-i) - 1|^2 = | -1 + 13i |^2 + | -1 - 13i |^2 = 1 + 169 + 1 + 169 = 340. Therefore, according to my previous logic, the answer should be S / 2 = 170, but in reality, the answer is 340. Therefore, my previous logic was wrong.

So where is the mistake?

Earlier, I thought that the sum over all roots of 1/|x|^2 is equal to S, and that the answer is S / 2. But in this case, the sum over all roots of 1/|x|^2 is 2 * (1/|r|^2) = 2 * 170 = 340, which is equal to S. Therefore, the answer is S, not S / 2. Therefore, my previous conclusion was wrong.

Wait, in this case, the sum over all roots of 1/|x|^2 is 340, which is equal to S. Therefore, the answer is 340, not S / 2. Therefore, my previous reasoning was incorrect.

So why was there a discrepancy? Let me re-examine.

Earlier, I thought that the sum over all roots of 1/|x|^2 is equal to S, and that the answer is half of that. But in reality, in this case, the answer is equal to S. Therefore, my mistake was in the assumption that the answer is half of S.

Let me re-examine the logic.

In the original problem, the sum we need is sum_{i=1}^5 1/(r_i \overline{r}_i). Each pair r_i and \overline{r}_i contributes one term, which is 1/(r_i \overline{r}_i) = 1/|r_i|^2. However, in the simple case, there are two roots, r and \overline{r}, and the sum is 1/(r \overline{r}) + 1/(\overline{r} r) = 2/(r \overline{r}) = 2/|r|^2. However, in this case, the sum over all roots of 1/|x|^2 is 2/|r|^2, which is equal to S. Therefore, the answer is equal to the sum over all roots of 1/|x|^2. Therefore, in the original problem, the answer is S, not S / 2.

Wait, but in the simple case, the answer is 340, which is equal to S. Therefore, my previous conclusion that the answer is S / 2 was wrong. Therefore, I need to re-examine the logic.

Let me go back.

In the original problem, the roots are r_1, \overline{r}_1, ..., r_5, \overline{r}_5. The sum we need is sum_{i=1}^5 1/(r_i \overline{r}_i). Each term 1/(r_i \overline{r}_i) is equal to 1/|r_i|^2. However, each pair r_i and \overline{r}_i corresponds to two roots, but the term 1/(r_i \overline{r}_i) is the same for both. Therefore, the sum over all roots of 1/|x|^2 is 2 * sum_{i=1}^5 1/|r_i|^2. Therefore, the answer is (sum over all roots of 1/|x|^2) / 2.

But in the simple case, the sum over all roots of 1/|x|^2 is 2/|r|^2, and the answer is 2/|r|^2, which would mean that the answer is equal to the sum over all roots of 1/|x|^2. Therefore, there is a contradiction.

Wait, in the simple case, the sum over all roots of 1/|x|^2 is 2/|r|^2, and the answer is also 2/|r|^2. Therefore, the answer is equal to the sum over all roots of 1/|x|^2. Therefore, my previous conclusion that the answer is S / 2 was wrong.

Wait, what is the relationship between the sum over all roots of 1/|x|^2 and the answer?

In the simple case, the answer is the sum over the pairs, which is 1/(r \overline{r}) + 1/(\overline{r} r) = 2/(r \overline{r}) = 2/|r|^2. But the sum over all roots of 1/|x|^2 is also 2/|r|^2. Therefore, the answer is equal to the sum over all roots of 1/|x|^2. Therefore, in the original problem, the answer is equal to S, not S / 2.

But why in the previous logic I thought that the answer is S / 2? Because I thought that the sum over all roots of 1/|x|^2 is twice the answer. But in reality, in the simple case, the answer is equal to the sum over all roots of 1/|x|^2.

Wait, let me clarify.

In the simple case, there are two roots: r and \overline{r}. The sum we need is 1/(r \overline{r}) + 1/(\overline{r} r) = 2/(r \overline{r}) = 2/|r|^2. However, the sum over all roots of 1/|x|^2 is also 2/|r|^2. Therefore, the answer is equal to the sum over all roots of 1/|x|^2.

Therefore, in the original problem, the answer is equal to the sum over all roots of 1/|x|^2, which is S = 1700. Therefore, the answer is 1700. But in the simple case, this would mean that the answer is 340, which matches.

But earlier, I thought that the answer was S / 2, but that was incorrect. Therefore, my mistake was in the previous logic.

So why was there confusion? Because I thought that the sum over the pairs is half of the total sum over all roots, but in reality, the sum over the pairs is equal to the total sum over all roots.

Wait, no. In the simple case, the sum over the pairs is 2/|r|^2, which is equal to the sum over all roots of 1/|x|^2. Therefore, the answer is equal to the sum over all roots of 1/|x|^2. Therefore, in the original problem, the answer is S = 1700.

But this contradicts the previous logic where I thought that the answer is S / 2. Where is the mistake?

Let me think again.

The problem says: the equation has 10 complex roots r1, \overline{r1}, ..., r5, \overline{r5}. Find the sum 1/(r1 \overline{r1}) + ... + 1/(r5 \overline{r5}).

Each term 1/(r_i \overline{r_i}) is equal to 1/|r_i|^2. Therefore, the sum is sum_{i=1}^5 1/|r_i|^2.

However, the sum over all roots of 1/|x|^2 is sum_{i=1}^{10} 1/|x_i|^2 = sum_{i=1}^5 [1/|r_i|^2 + 1/|\overline{r_i}|^2] = sum_{i=1}^5 2/|r_i|^2. Therefore, the sum over all roots is twice the desired sum. Therefore, the desired sum is (sum over all roots of 1/|x|^2) / 2.

But in the simple case, the sum over all roots of 1/|x|^2 is 2/|r|^2, and the desired sum is also 2/|r|^2. Therefore, this suggests that (sum over all roots of 1/|x|^2) / 2 = 2/|r|^2 / 2 = 1/|r|^2, which contradicts.

Wait, this is confusing. Let me clarify with the simple case.

In the simple case, the roots are r and \overline{r}. The desired sum is 1/(r \overline{r}) + 1/(\overline{r} r) = 2/(r \overline{r}) = 2/|r|^2.

The sum over all roots of 1/|x|^2 is 1/|r|^2 + 1/|\overline{r}|^2 = 2/|r|^2.

Therefore, the desired sum is equal to the sum over all roots of 1/|x|^2.

But according to the previous logic, the desired sum should be (sum over all roots of 1/|x|^2) / 2. But in reality, it is equal to the sum over all roots of 1/|x|^2.

This suggests that my previous logic was wrong. Where is the mistake?

Ah! The problem says that the roots are r1, \overline{r1}, ..., r5, \overline{r5}. Therefore, the sum we need is sum_{i=1}^5 1/(r_i \overline{r_i}) = sum_{i=1}^5 1/|r_i|^2. However, the sum over all roots of 1/|x|^2 is sum_{i=1}^{10} 1/|x_i|^2 = sum_{i=1}^5 [1/|r_i|^2 + 1/|r_i|^2] = 2 * sum_{i=1}^5 1/|r_i|^2. Therefore, the desired sum is (sum over all roots of 1/|x|^2) / 2.

But in the simple case, this would mean that the desired sum is (2/|r|^2) / 2 = 1/|r|^2, but in reality, the desired sum is 2/|r|^2. Therefore, there is a contradiction.

Wait, what is the desired sum in the simple case? The problem says: "the equation has 10 complex roots r1, \overline{r1}, ..., r5, \overline{r5}, where the bar denotes complex conjugation. Find the value of ...". In the simple case, the equation is degree 2, so it has 2 roots, which are r1 and \overline{r1}. Therefore, the sum is 1/(r1 \overline{r1}) = 1/|r1|^2. But in our calculation, the sum over all roots of 1/|x|^2 is 2/|r1|^2, and the desired sum is 1/|r1|^2, which is half of the total sum. Therefore, in this case, the desired sum is (sum over all roots of 1/|x|^2) / 2.

But earlier, I thought that the desired sum was 2/|r|^2, but that was incorrect. Let me re-examine the simple case.

In the simple case, the equation is x^2 + (13x - 1)^2 = 0, which has two roots: r and \overline{r}. The problem would say that the equation has 2 complex roots r1, \overline{r1}, and the sum is 1/(r1 \overline{r1}). Therefore, the desired sum is 1/|r1|^2. However, in our calculation, the sum over all roots of 1/|x|^2 is 2/|r1|^2, so the desired sum is half of that.

But earlier, I thought that the desired sum was 2/|r|^2, but that was a mistake. The correct desired sum is 1/|r|^2, which is half of the total sum over all roots.

Therefore, in the original problem, the desired sum is (sum over all roots of 1/|x|^2) / 2 = S / 2 = 1700 / 2 = 850.

But why did I get confused earlier? Because I incorrectly calculated the desired sum as 2/|r|^2, but actually, it's 1/|r|^2.

Let me verify with the simple case.

In the simple case, the roots are r and \overline{r}. The problem says to find 1/(r \overline{r}) + ... but since there is only one pair, it's just 1/(r \overline{r}) = 1/|r|^2. The sum over all roots of 1/|x|^2 is 1/|r|^2 + 1/|\overline{r}|^2 = 2/|r|^2. Therefore, the desired sum is half of that, which is 1/|r|^2.

Therefore, in the original problem, the desired sum is S / 2 = 1700 / 2 = 850.

But in the simple case, S = 340, and the desired sum is 170, which is S / 2. But earlier, I thought that the desired sum was 340, but that was incorrect.

Wait, let me recalculate the simple case.

In the simple case, the equation is x^2 + (13x - 1)^2 = 0. The roots are x = [13 ± i]/170. Therefore, |x|^2 = (13^2 + 1^2)/170^2 = 170 / 28900 = 1/170. Therefore, 1/|x|^2 = 170. Therefore, the sum over all roots of 1/|x|^2 is 170 + 170 = 340. Therefore, the desired sum is 340 / 2 = 170.

But earlier, I thought that the desired sum was 2/|r|^2, but that was incorrect. The correct desired sum is 170, which is S / 2.

Therefore, my initial conclusion was correct, and the confusion was due to miscalculating the desired sum in the simple case.

Therefore, returning to the original problem, the answer is S / 2 = 1700 / 2 = 850.

Therefore, the correct answer is 850.

But to confirm, let me check with the simple case. If the answer is 170, which is S / 2 = 340 / 2 = 170, and in reality, 1/|x|^2 for each root is 170, so the sum over all roots is 340, and the desired sum is 170.

Therefore, the logic holds.

Therefore, the answer is 850.

**Final Answer**
\boxed{850}

To solve the equation

$$
x^{10} + (13x - 1)^{10} = 0,
$$

we are given that it has 10 complex roots, which come in conjugate pairs: $ r_1, \overline{r}_1, r_2, \overline{r}_2, \ldots, r_5, \overline{r}_5 $. The goal is to compute the sum:

$$
\frac{1}{r_1\overline{r}_1} + \frac{1}{r_2\overline{r}_2} + \cdots + \frac{1}{r_5\overline{r}_5}.
$$

---

### Step 1: Understanding the Expression

For any complex number $ z $, the product $ z\overline{z} = |z|^2 $, the square of its modulus. Therefore, the expression becomes:

$$
\sum_{i=1}^5 \frac{1}{r_i \overline{r}_i} = \sum_{i=1}^5 \frac{1}{|r_i|^2}.
$$

So, we are to compute the sum of the reciprocals of the squared moduli of the roots.

---

### Step 2: Rewriting the Equation

We can rewrite the original equation as:

$$
x^{10} = - (13x - 1)^{10}.
$$

Taking both sides to the power of $ \frac{1}{10} $, we get:

$$
\left( \frac{x}{13x - 1} \right)^{10} = -1.
$$

Let $ y = \frac{x}{13x - 1} $. Then $ y^{10} = -1 $, so $ y $ is a 10th root of $-1$. The 10th roots of $-1$ are the complex numbers $ y = e^{i(\pi + 2\pi k)/10} $ for $ k = 0, 1, \ldots, 9 $, which are all on the unit circle.

Solving for $ x $ in terms of $ y $, we get:

$$
x = \frac{y}{13y - 1}.
$$

So each root $ x $ corresponds to a root $ y $ of $ y^{10} = -1 $.

---

### Step 3: Expressing the Desired Sum

We want to compute:

$$
\sum_{i=1}^5 \frac{1}{|r_i|^2} = \sum_{i=1}^5 \frac{1}{|x_i|^2}.
$$

From the expression $ x = \frac{y}{13y - 1} $, we can compute:

$$
|x|^2 = \left| \frac{y}{13y - 1} \right|^2 = \frac{|y|^2}{|13y - 1|^2}.
$$

Since $ |y| = 1 $ (because $ y $ lies on the unit circle), this simplifies to:

$$
|x|^2 = \frac{1}{|13y - 1|^2} \quad \Rightarrow \quad \frac{1}{|x|^2} = |13y - 1|^2.
$$

Thus, the sum becomes:

$$
\sum_{i=1}^5 \frac{1}{|r_i|^2} = \sum_{i=1}^5 |13y_i - 1|^2,
$$

where $ y_i $ are the 10th roots of $-1$, and we are summing over 5 distinct pairs of conjugate roots.

However, since the roots $ y $ come in conjugate pairs, and $ |13y - 1|^2 = |13\overline{y} - 1|^2 $, each pair contributes the same value. Therefore, the total sum over all 10 roots is:

$$
\sum_{y^{10} = -1} |13y - 1|^2 = 2 \sum_{i=1}^5 |13y_i - 1|^2.
$$

So, the desired sum is:

$$
\sum_{i=1}^5 \frac{1}{|r_i|^2} = \frac{1}{2} \sum_{y^{10} = -1} |13y - 1|^2.
$$

---

### Step 4: Computing the Sum

We now compute:

$$
S = \sum_{y^{10} = -1} |13y - 1|^2.
$$

Using the identity $ |a - b|^2 = (a - b)(\overline{a} - \overline{b}) $, and noting that $ |y| = 1 $, we have:

$$
|13y - 1|^2 = (13y - 1)(13\overline{y} - 1) = 169 - 13y - 13\overline{y} + 1 = 170 - 13(y + \overline{y}).
$$

Since $ y $ are the 10th roots of $-1$, and $ y^{10} = -1 $, the sum of all such $ y $ is zero (by Vieta's formula, since the polynomial $ y^{10} + 1 = 0 $ has zero coefficient for $ y^9 $). Similarly, the sum of $ \overline{y} $ is also zero. Therefore:

$$
\sum_{y^{10} = -1} |13y - 1|^2 = \sum_{y^{10} = -1} (170 - 13(y + \overline{y})) = 10 \cdot 170 - 13 \cdot 0 = 1700.
$$

Thus, the desired sum is:

$$
\frac{1}{2} \cdot 1700 = \boxed{850}.
$$

---

### Final Answer

$$
\boxed{850}
$$

[Assistant]
Explanation:
Okay, so I need to solve this equation: $x^{10} + (13x - 1)^{10} = 0$. The problem says it has 10 complex roots, which are paired as conjugates: $r_1, \overline{r}_1, r_2, \overline{r}_2,$ and so on up to $r_5, \overline{r}_5$. Then I need to find the sum $\frac{1}{r_1\overline{r}_1} + \frac{1}{r_2\overline{r}_2} + \cdots + \frac{1}{r_5\overline{r}_5}$.

First, let me recall that for any complex number $z$, the product $z \overline{z}$ is equal to $|z|^2$, the square of its modulus. So each term in the sum is $1/|r_i|^2$. Therefore, the problem is asking for the sum of the reciprocals of the squared moduli of the roots.

But how do I find that? Maybe I can relate this to the coefficients of the polynomial? Since the polynomial is given, perhaps I can use Vieta's formulas or some other properties of polynomials?

Let me first try to understand the equation better. The equation is $x^{10} + (13x - 1)^{10} = 0$. Let me rewrite this as:

$$
x^{10} = - (13x - 1)^{10}
$$

Taking both sides to the power of 1/10, but since we are dealing with complex numbers, there are multiple roots. However, maybe I can write this as:

$$
\left( \frac{x}{13x - 1} \right)^{10} = -1
$$

So, $\frac{x}{13x - 1}$ is a 10th root of -1. Let me denote $y = \frac{x}{13x - 1}$. Then $y^{10} = -1$. Therefore, $y$ is a complex number such that $y^{10} = -1$. The solutions to this equation are the 10th roots of -1, which can be written as $y = e^{i(\pi + 2\pi k)/10}$ for $k = 0, 1, ..., 9$.

Alternatively, since $-1 = e^{i\pi}$, the 10th roots of -1 are $e^{i(\pi + 2\pi k)/10}$ for $k = 0$ to 9. So each $y$ is a complex number on the unit circle at angles $\pi/10, 3\pi/10, ..., 19\pi/10$.

But maybe there's a better way to approach this. Let me try to express $x$ in terms of $y$. From $y = \frac{x}{13x - 1}$, solving for $x$:

Multiply both sides by $13x - 1$:

$$
y(13x - 1) = x \\
13y x - y = x \\
13y x - x = y \\
x(13y - 1) = y \\
x = \frac{y}{13y - 1}
$$

So each root $x$ of the original equation can be written as $x = \frac{y}{13y - 1}$, where $y^{10} = -1$. Therefore, the roots $x$ are given by this expression for each 10th root of -1.

But since the original equation is degree 10, there are 10 roots, which matches with the 10 values of $y$. However, the problem mentions that the roots come in conjugate pairs. So perhaps some of these $y$ values are conjugates of each other? Let me check.

If $y$ is a root of $y^{10} = -1$, then its complex conjugate $\overline{y}$ is also a root because if $y^{10} = -1$, then $\overline{y}^{10} = \overline{(-1)} = -1$. Therefore, the roots $y$ come in conjugate pairs. Therefore, when we take $x = \frac{y}{13y - 1}$, then for each pair $y$ and $\overline{y}$, we get two roots $x$ and $\overline{x}$, since substituting $\overline{y}$ into the expression would give $\overline{x}$. Therefore, the roots $x$ also come in conjugate pairs, as stated in the problem.

Therefore, each pair $r_i$ and $\overline{r}_i$ corresponds to a pair of $y$ values $y$ and $\overline{y}$. Therefore, perhaps I can compute $1/(r_i \overline{r}_i)$ for each pair and sum them up.

But how do I compute $1/(r_i \overline{r}_i)$? Since $r_i \overline{r}_i = |r_i|^2$, so $1/|r_i|^2$. Alternatively, since $x = \frac{y}{13y - 1}$, then $|x|^2 = \frac{|y|^2}{|13y - 1|^2}$. Therefore, $1/|x|^2 = \frac{|13y - 1|^2}{|y|^2}$. Therefore, for each root $x$ corresponding to a $y$, $1/|x|^2 = \frac{|13y - 1|^2}{|y|^2}$.

But since $y$ is on the unit circle (because $y^{10} = -1$, so $|y| = 1$), then $|y|^2 = 1$. Therefore, $1/|x|^2 = |13y - 1|^2$. Therefore, the sum we need is the sum over all pairs $r_i, \overline{r}_i$ of $|13y - 1|^2$, but since each pair corresponds to a $y$ and $\overline{y}$, and since $|13y - 1|^2 = |13\overline{y} - 1|^2$ (because modulus squared is invariant under conjugation), then each pair contributes twice the value of $|13y - 1|^2$? Wait, no. Wait, each pair $y$ and $\overline{y}$ corresponds to two roots $x$ and $\overline{x}$, so each pair contributes two terms in the sum: $1/|x|^2$ and $1/|\overline{x}|^2$, but since $1/|\overline{x}|^2 = 1/|x|^2$, so each pair contributes twice $1/|x|^2$. However, since we have 5 pairs, the total sum is 2 times the sum over each pair's $1/|x|^2$. Wait, but actually, the original sum is over 5 terms, each being $1/(r_i \overline{r}_i)$. Since each pair $r_i$ and $\overline{r}_i$ contributes one term $1/(r_i \overline{r}_i)$, which is equal to $1/|r_i|^2$. Therefore, since each pair corresponds to two roots $x$ and $\overline{x}$, but the term $1/|x|^2$ is the same for both, so each pair contributes one term. Therefore, the total sum is the sum over all 5 pairs of $1/|x|^2$, which is equal to the sum over all 10 roots of $1/|x|^2$ divided by 2? Wait, no. Wait, each pair contributes one term, so the total sum is the sum over each pair's $1/|x|^2$. But since each pair corresponds to two roots, but the term is the same for both, so the total sum is the sum over all roots of $1/|x|^2$ divided by 2? Wait, no. Let me think again.

Suppose there are 10 roots, each paired with their conjugate. So for each pair $r_i$ and $\overline{r}_i$, we have two roots. The sum given in the problem is over 5 terms, each being $1/(r_i \overline{r}_i)$. Since $1/(r_i \overline{r}_i) = 1/|r_i|^2$, and since $1/|r_i|^2 = 1/| \overline{r}_i |^2$, so each pair contributes one term. Therefore, the total sum is the sum over all 5 pairs of $1/|r_i|^2$. Therefore, if I can compute $1/|x|^2$ for each root $x$, and then sum them up, but since each pair contributes the same value twice? Wait, no. Wait, if I have two roots $x$ and $\overline{x}$, then $1/|x|^2$ and $1/|\overline{x}|^2$ are the same, so each pair contributes twice the same value? Wait, no. Wait, the problem says the sum is over 5 terms, each being $1/(r_i \overline{r}_i)$. So each pair contributes one term. Therefore, if I can compute $1/|x|^2$ for each root, and then sum them up, but since each pair is counted once, then the total sum is the sum over all roots of $1/|x|^2$ divided by 2? Wait, no. Let me think with an example. Suppose there are two roots, $r$ and $\overline{r}$. Then the sum would be $1/(r \overline{r}) + 1/(\overline{r} r) = 2/(r \overline{r})$. But in the problem, the sum is over 5 terms, each being $1/(r_i \overline{r}_i)$. Therefore, if there are 5 pairs, then the total sum is 5 terms, each being $1/(r_i \overline{r}_i)$. Therefore, if I have 10 roots, each pair contributes one term. Therefore, the total sum is equal to the sum over all roots of $1/|x|^2$ divided by 2? Wait, no. If each pair contributes one term, then the total sum is equal to the sum over all roots of $1/|x|^2$ divided by 2? Wait, no. Let me think again. Suppose there are two roots, $r$ and $\overline{r}$. Then the sum is $1/(r \overline{r})$, which is $1/|r|^2$. However, if I compute $1/|r|^2 + 1/|\overline{r}|^2$, that would be $2/|r|^2$. Therefore, if the problem is asking for the sum over the pairs, which is 1 term per pair, then it's $1/|r|^2$, but if I compute the sum over all roots, it would be $2/|r|^2$. Therefore, in our case, the problem is asking for the sum over 5 terms, each being $1/|r_i|^2$, so that would be half of the sum over all roots of $1/|x|^2$. Therefore, if I can compute the sum over all roots of $1/|x|^2$, then divide by 2, I would get the desired value.

But how do I compute the sum over all roots of $1/|x|^2$? Alternatively, since $1/|x|^2 = |1/x|^2$, so the sum is the sum over all roots of $|1/x|^2$. Therefore, if I can compute the sum of $|1/x|^2$ over all roots, then divide by 2, I get the answer.

Alternatively, since $x$ satisfies the equation $x^{10} + (13x - 1)^{10} = 0$, which can be rewritten as $(13x - 1)^{10} = -x^{10}$. Taking modulus on both sides, we get $|13x - 1|^{10} = |x|^{10}$. Therefore, $|13x - 1| = |x|$. Therefore, $|13x - 1|^2 = |x|^2$. Therefore, $|13x - 1|^2 = |x|^2$. Wait, but earlier I had that $1/|x|^2 = |13y - 1|^2$, where $y = x/(13x - 1)$. But if $|13x - 1|^2 = |x|^2$, then $|13x - 1|^2 / |x|^2 = 1$, so $1/|x|^2 = |13x - 1|^2 / |x|^2 * |x|^2 / |x|^2$... Wait, maybe I need to check again.

Wait, earlier I had that $x = y/(13y - 1)$, and since $y^{10} = -1$, and $|y| = 1$. Then $|x|^2 = |y|^2 / |13y - 1|^2 = 1 / |13y - 1|^2$. Therefore, $1/|x|^2 = |13y - 1|^2$. Therefore, the sum over all roots of $1/|x|^2$ is equal to the sum over all $y$ (which are the 10th roots of -1) of $|13y - 1|^2$. Therefore, since each root $x$ corresponds to a unique $y$, and there are 10 roots, the sum over all roots of $1/|x|^2$ is equal to the sum over all $y$ of $|13y - 1|^2$. Therefore, the desired sum (which is half of that) is equal to half of the sum over all $y$ of $|13y - 1|^2$.

Therefore, if I can compute the sum over all 10th roots of -1 of $|13y - 1|^2$, then divide by 2, I get the answer.

So let me compute that sum. Let me denote $S = \sum_{y^{10} = -1} |13y - 1|^2$. Then the answer is $S / 2$.

Now, how do I compute $S$? Let me recall that for any complex number $z$, $|z|^2 = z \overline{z}$. Therefore, $|13y - 1|^2 = (13y - 1)(13\overline{y} - 1)$. Since $y^{10} = -1$, then $\overline{y} = y^{-10} \cdot (-1)^{-1}$. Wait, maybe not. Wait, since $y^{10} = -1$, then taking modulus squared on both sides, $|y|^{10} = |-1| = 1$, so $|y| = 1$. Therefore, $\overline{y} = 1/y$. Therefore, $|13y - 1|^2 = (13y - 1)(13\overline{y} - 1) = (13y - 1)(13/y - 1)$. Let me compute that:

$$
(13y - 1)\left( \frac{13}{y} - 1 \right) = 13y \cdot \frac{13}{y} - 13y \cdot 1 - 1 \cdot \frac{13}{y} + 1 \cdot 1 = 169 - 13y - \frac{13}{y} + 1 = 170 - 13\left(y + \frac{1}{y}\right)
$$

Therefore, $|13y - 1|^2 = 170 - 13\left(y + \frac{1}{y}\right)$. Therefore, the sum $S$ becomes:

$$
S = \sum_{y^{10} = -1} \left[170 - 13\left(y + \frac{1}{y}\right)\right] = 10 \times 170 - 13 \sum_{y^{10} = -1} \left(y + \frac{1}{y}\right)
$$

So, $S = 1700 - 13 \times \left[ \sum_{y^{10} = -1} y + \sum_{y^{10} = -1} \frac{1}{y} \right]$.

But since $y$ are the 10th roots of -1, which are the same as the 20th roots of 1 that are not 10th roots of 1. Wait, maybe not. Let me think. The equation $y^{10} = -1$ can be rewritten as $y^{20} = 1$, so the solutions are the 20th roots of unity, but only those that satisfy $y^{10} = -1$, which are the primitive 20th roots of unity and the ones with order dividing 20 but not dividing 10. Wait, more precisely, the solutions to $y^{10} = -1$ are the 20th roots of unity that are not 10th roots of unity. Because if $y^{10} = -1$, then $y^{20} = 1$, so they are 20th roots of unity. However, the 10th roots of unity satisfy $y^{10} = 1$, so the solutions to $y^{10} = -1$ are the 20th roots of unity that are not 10th roots of unity. Therefore, there are 10 such roots.

Alternatively, since $y^{10} = -1$, we can write $y = e^{i\theta}$, where $10\theta = \pi + 2\pi k$, so $\theta = \pi/10 + \pi k/5$, for $k = 0, 1, ..., 9$. Therefore, the roots are $e^{i(\pi/10 + \pi k/5)}$ for $k = 0$ to 9. Let me check for k=0: $\pi/10$, k=1: $\pi/10 + \pi/5 = 3\pi/10$, k=2: $\pi/10 + 2\pi/5 = 5\pi/10 = \pi/2$, k=3: $\pi/10 + 3\pi/5 = 7\pi/10$, k=4: $\pi/10 + 4\pi/5 = 9\pi/10$, k=5: $\pi/10 + \pi = 11\pi/10$, k=6: $\pi/10 + 6\pi/5 = 13\pi/10$, k=7: $\pi/10 + 7\pi/5 = 15\pi/10 = 3\pi/2$, k=8: $\pi/10 + 8\pi/5 = 17\pi/10$, k=9: $\pi/10 + 9\pi/5 = 19\pi/10$. So these are the angles for the roots.

But maybe I don't need to go into angles. Let me think about the sum $\sum_{y^{10} = -1} y$. Since the roots of $y^{10} + 1 = 0$ are the roots of $y^{10} = -1$, which is a polynomial of degree 10. Therefore, the sum of the roots is given by Vieta's formula. The polynomial is $y^{10} + 1 = 0$, which can be written as $y^{10} + 0y^9 + 0y^8 + \cdots + 0y + 1 = 0$. Therefore, the sum of the roots is $-0/1 = 0$. Therefore, $\sum_{y^{10} = -1} y = 0$. Similarly, $\sum_{y^{10} = -1} \frac{1}{y}$. Since $y$ are roots of $y^{10} + 1 = 0$, then $1/y$ are roots of $1 + y^{10} = 0$, which is the same equation. Therefore, the sum $\sum \frac{1}{y}$ is also equal to the sum of the roots of $y^{10} + 1 = 0$, which is 0. Therefore, both sums are zero. Therefore, $S = 1700 - 13 \times (0 + 0) = 1700$. Therefore, the sum over all roots of $1/|x|^2$ is 1700, so the desired sum is 1700 / 2 = 850. Wait, but hold on. Wait, the problem says the answer is the sum over 5 terms, each being $1/(r_i \overline{r}_i)$. But according to this, the sum over all roots of $1/|x|^2$ is 1700, so the sum over the pairs would be 1700 / 2 = 850. Therefore, the answer is 850? But let me check again.

Wait, earlier I said that the sum over all roots of $1/|x|^2$ is equal to S = 1700, and since each pair contributes one term, then the answer is 1700 / 2 = 850. But let me check again the logic.

Each root x corresponds to a y, and for each pair of roots x and \overline{x}, they correspond to y and \overline{y}. Therefore, the sum over all roots of 1/|x|^2 is equal to the sum over all y of |13y - 1|^2, which is S = 1700. However, since each pair (x, \overline{x}) contributes two roots, but the sum over all roots includes both. However, the problem asks for the sum over 5 terms, each being 1/(r_i \overline{r}_i). Since each pair contributes one term, which is 1/|r_i|^2, and since there are 5 pairs, the total sum is 5 terms. However, the sum over all roots of 1/|x|^2 is 2 times the sum over the pairs. Because each pair contributes two roots, each with the same 1/|x|^2. Therefore, if the total sum over all roots is 1700, then the sum over the pairs is 1700 / 2 = 850. Therefore, the answer is 850.

But let me check with a simpler case. Suppose I have two roots, r and \overline{r}. Then the sum over all roots of 1/|x|^2 is 1/|r|^2 + 1/|\overline{r}|^2 = 2/|r|^2. The sum over the pairs is just 1/|r|^2. Therefore, yes, the sum over the pairs is half of the total sum over all roots. Therefore, in our case, since there are 10 roots, the sum over the pairs is 1700 / 2 = 850. Therefore, the answer is 850.

But let me check if my calculation of S is correct. Let me recap:

We had S = sum_{y^{10} = -1} |13y - 1|^2 = 1700 - 13*(sum y + sum 1/y). Then since sum y = 0 and sum 1/y = 0, so S = 1700.

But why is sum y = 0? Because the polynomial is y^{10} + 1 = 0, which is y^{10} + 0y^9 + ... + 0y + 1 = 0. Therefore, the sum of roots is -coefficient of y^9 / coefficient of y^{10} = -0/1 = 0. Similarly, sum 1/y is the same as sum of roots of the polynomial obtained by reversing coefficients. The polynomial is y^{10} + 1 = 0. Reversing coefficients gives 1 + y^{10} = 0, which is the same equation. Therefore, the sum of 1/y is also 0. Therefore, yes, S = 1700.

Therefore, the answer is 850. However, let me check with another approach to make sure.

Alternative approach: Let me consider the original equation $x^{10} + (13x - 1)^{10} = 0$. Let me divide both sides by $x^{10}$ (assuming x ≠ 0, which is valid since if x=0, then 0 + (-1)^10 = 1 ≠ 0, so x=0 is not a root). Therefore, we get:

$$
1 + \left(13 - \frac{1}{x}\right)^{10} = 0
$$

Therefore,

$$
\left(13 - \frac{1}{x}\right)^{10} = -1
$$

Let me denote $z = 13 - \frac{1}{x}$. Then $z^{10} = -1$, so z is a 10th root of -1, same as before. Therefore, $z = e^{i(\pi + 2\pi k)/10}$ for k=0,...,9. Then solving for x:

$$
z = 13 - \frac{1}{x} \implies \frac{1}{x} = 13 - z \implies x = \frac{1}{13 - z}
$$

Therefore, each root x is given by $x = 1/(13 - z)$, where z^{10} = -1. Therefore, similar to before, but with a different expression.

Now, let me compute $1/|x|^2 = |13 - z|^2$. Therefore, the sum over all roots of $1/|x|^2$ is equal to sum_{z^{10}=-1} |13 - z|^2. Which is the same as before, since earlier we had |13y - 1|^2, but here it's |13 - z|^2. However, since z is a root of z^{10} = -1, which is same as y^{10} = -1. Therefore, the sum is same as before. Therefore, same result. Therefore, S = 1700. Therefore, the answer is 850.

But let me check with specific values. Let me take a simple case. Suppose I take a root z such that z^{10} = -1. Let me pick z = e^{i\pi/10}. Then compute |13 - z|^2. Similarly for other roots. But since the sum over all roots is 1700, which is 10*170, and since each term is 170 - 13*(z + 1/z), and sum over z + 1/z is zero. Therefore, the total sum is 10*170 = 1700. Therefore, seems correct.

Alternatively, maybe I can compute the sum using another method. Let me recall that for any complex number z, |a - z|^2 = |a|^2 + |z|^2 - 2 Re(a \overline{z}). Therefore, |13 - z|^2 = 13^2 + |z|^2 - 2 Re(13 \overline{z}). Since |z| = 1 (because z^{10} = -1 implies |z|^{10} = 1, so |z| = 1). Therefore, |13 - z|^2 = 169 + 1 - 2 Re(13 \overline{z}) = 170 - 26 Re(\overline{z}) = 170 - 26 Re(z). Because Re(\overline{z}) = Re(z). Therefore, sum_{z^{10}=-1} |13 - z|^2 = sum_{z} [170 - 26 Re(z)] = 10*170 - 26 sum Re(z).

But sum Re(z) over all z^{10} = -1 is equal to Re(sum z) = Re(0) = 0. Therefore, the total sum is 1700 - 0 = 1700. Therefore, same result. Therefore, this confirms that S = 1700. Therefore, the answer is 850.

But let me check once more. Suppose I take a specific root. Let me take z = e^{i\pi/10}. Then compute |13 - z|^2. But even if I compute it, the sum over all roots would still be 1700. Therefore, seems correct.

Therefore, the answer is 850. However, let me check if there is a mistake in the logic.

Wait, the problem says that the roots are r1, \overline{r1}, ..., r5, \overline{r5}. Therefore, there are 10 roots. The sum we need is sum_{i=1}^5 1/(r_i \overline{r_i}) = sum_{i=1}^5 1/|r_i|^2. But since each pair contributes two roots, but the term is same for both, so the total sum is sum_{i=1}^5 1/|r_i|^2. However, the sum over all roots of 1/|x|^2 is 2 * sum_{i=1}^5 1/|r_i|^2. Therefore, the answer is S / 2 = 1700 / 2 = 850. Therefore, yes.

Alternatively, maybe I can think in terms of polynomials. Let me consider the original equation: x^{10} + (13x - 1)^{10} = 0. Let me denote this polynomial as P(x) = x^{10} + (13x - 1)^{10}. Let me consider the polynomial Q(x) = P(x). The roots of Q(x) are the roots we are considering.

We need to compute sum_{i=1}^5 1/(r_i \overline{r_i}) = sum_{i=1}^5 1/|r_i|^2.

But for a polynomial with roots r_i, the sum of 1/|r_i|^2 can be related to the coefficients? Let me think.

Note that 1/|r_i|^2 = |1/r_i|^2. Therefore, if I consider the polynomial whose roots are 1/r_i, then the sum of |1/r_i|^2 is the sum of the squares of the moduli of the roots of that polynomial.

But how to find that polynomial? Let me try. Let me denote y = 1/x. Then x = 1/y. Substitute into P(x) = 0:

(1/y)^{10} + (13*(1/y) - 1)^{10} = 0

Multiply both sides by y^{10}:

1 + (13 - y)^{10} = 0

Therefore, the polynomial equation for y is (13 - y)^{10} + 1 = 0, which is same as (y - 13)^{10} + 1 = 0. Therefore, the roots y_i = 1/r_i satisfy (y_i - 13)^{10} = -1. Therefore, similar to before. Therefore, the roots y_i are 13 - z, where z^{10} = -1. Therefore, same as before. Therefore, the roots y_i are 13 - z, where z^{10} = -1. Therefore, the sum of |y_i|^2 is sum |13 - z|^2, which is S = 1700. Therefore, the sum of |y_i|^2 is 1700. However, the sum we need is sum_{i=1}^5 1/|r_i|^2 = sum_{i=1}^5 |y_i|^2. Wait, because 1/|r_i|^2 = |y_i|^2. Therefore, the sum we need is sum |y_i|^2. However, the sum over all roots y_i of |y_i|^2 is 1700. But since there are 10 roots y_i, and they come in conjugate pairs, then the sum over all roots of |y_i|^2 is 2 * sum_{i=1}^5 |y_i|^2. Therefore, the sum we need is 1700 / 2 = 850. Therefore, same result.

Therefore, this confirms again that the answer is 850.

But let me check with Vieta's formula. Suppose I want to compute sum |y_i|^2. For complex numbers, |y_i|^2 = y_i \overline{y_i}. Therefore, sum |y_i|^2 = sum y_i \overline{y_i}. However, since the roots y_i are 13 - z, where z^{10} = -1. Therefore, the roots y_i are 13 - z, and their conjugates are 13 - \overline{z}. Therefore, for each pair y and \overline{y}, we have y = 13 - z and \overline{y} = 13 - \overline{z}. Therefore, the product y \overline{y} = |y|^2 = |13 - z|^2. Therefore, the sum over all roots of |y|^2 is sum |13 - z|^2 = 1700. Therefore, same as before.

Alternatively, if I consider the polynomial whose roots are y_i = 13 - z, where z^{10} = -1. Let me find the polynomial for y. Let me denote z^{10} = -1. Then, z^{10} + 1 = 0. Let me express z in terms of y: z = 13 - y. Therefore, substituting into z^{10} + 1 = 0:

(13 - y)^{10} + 1 = 0. Therefore, the polynomial is (y - 13)^{10} + 1 = 0. Therefore, the roots are y_i = 13 - z, as before.

Now, to compute sum |y_i|^2, which is sum y_i \overline{y_i}. However, since the polynomial has real coefficients? Wait, the polynomial (y - 13)^{10} + 1 = 0 has real coefficients. Therefore, the roots come in conjugate pairs. Therefore, the sum of y_i \overline{y_i} is equal to sum |y_i|^2.

But how can I compute this sum? For a polynomial with real coefficients, the sum of |y_i|^2 can be related to the coefficients. Let me recall that for a polynomial with roots y_i, sum |y_i|^2 = sum y_i \overline{y_i} = sum y_i \overline{y_i}. However, since the polynomial has real coefficients, the non-real roots come in conjugate pairs. Therefore, for each pair y and \overline{y}, their contribution to the sum is y \overline{y} + \overline{y} y = 2 |y|^2. Wait, no. Wait, if you have a pair y and \overline{y}, then the sum over their |y|^2 is |y|^2 + |\overline{y}|^2 = 2 |y|^2. However, if you have a real root, then |y|^2 = y^2. Therefore, the total sum is sum_{i=1}^{10} |y_i|^2.

But how to compute this sum? Let me recall that for any complex number, |y|^2 = y \overline{y}. Therefore, sum |y_i|^2 = sum y_i \overline{y_i}. However, since the polynomial has real coefficients, the sum over y_i \overline{y_i} can be related to the coefficients.

Alternatively, note that sum |y_i|^2 = sum y_i \overline{y_i} = sum y_i \overline{y_i} = sum y_i \overline{y_i}. But since the polynomial is (y - 13)^{10} + 1 = 0, which is a polynomial with real coefficients. Let me denote the polynomial as P(y) = (y - 13)^{10} + 1. Let me expand this polynomial. However, expanding it might be complicated, but maybe I can use Vieta's formula.

Alternatively, note that sum |y_i|^2 = sum y_i \overline{y_i} = sum y_i \overline{y_i} = sum y_i \overline{y_i}. However, since the coefficients are real, the sum over y_i \overline{y_i} is equal to the sum over y_i \overline{y_i} = sum y_i \overline{y_i} = sum y_i \overline{y_i}.

Alternatively, consider that for any polynomial with real coefficients, the sum of the squares of the moduli of the roots can be calculated using the formula:

sum |y_i|^2 = (sum y_i^2) + (sum \overline{y_i}^2) - 2 Re(sum y_i \overline{y_i})? Wait, no. Wait, maybe not. Let me think.

Alternatively, note that for any complex number, |y|^2 = y \overline{y}. Therefore, sum |y_i|^2 = sum y_i \overline{y_i}. However, since the polynomial has real coefficients, the sum over y_i \overline{y_i} is equal to the sum over y_i \overline{y_i} = sum y_i \overline{y_i}. But how to compute this?

Alternatively, consider that if we have a polynomial with roots y_i, then the sum of y_i \overline{y_i} can be related to the coefficients. However, since the polynomial has real coefficients, the sum over y_i \overline{y_i} is equal to the sum over y_i \overline{y_i} = sum y_i \overline{y_i}. But this seems circular.

Alternatively, consider that if we have a polynomial with real coefficients, then the sum over y_i \overline{y_i} is equal to the sum over y_i \overline{y_i} = sum y_i \overline{y_i} = sum y_i \overline{y_i}.

Alternatively, think of the polynomial whose roots are y_i and \overline{y_i}. But since the polynomial already has real coefficients, the roots come in conjugate pairs. Therefore, the sum over y_i \overline{y_i} is equal to the sum over |y_i|^2.

But perhaps I can use the fact that for a polynomial P(y) = \prod_{i=1}^{n} (y - y_i), then the sum of y_i \overline{y_i} can be related to the coefficients. However, this seems complicated.

Alternatively, since we already have that sum |y_i|^2 = 1700, from previous calculations, and that this is equal to the sum over all roots of |13 - z|^2, which we calculated as 1700, then we can trust that result. Therefore, the answer is 850.

But to be thorough, let me check with a small example. Let me take a simpler equation where I can compute manually. For example, suppose the equation is x^2 + (13x - 1)^2 = 0. Then, solving this would give x^2 + (13x - 1)^2 = 0 => x^2 + 169x^2 - 26x + 1 = 0 => 170x^2 - 26x + 1 = 0. The roots are [26 ± sqrt(676 - 680)] / 340. Wait, discriminant is 676 - 680 = -4. Therefore, roots are [26 ± 2i]/340 = [13 ± i]/170. Therefore, the roots are (13 + i)/170 and (13 - i)/170. Therefore, their moduli squared are [(13)^2 + 1^2]/(170^2) = (169 + 1)/28900 = 170/28900 = 1/170. Therefore, the sum of 1/|r_i|^2 is 1/(1/170) + 1/(1/170) = 170 + 170 = 340. However, according to our formula, S would be sum |13y - 1|^2 where y^2 = -1. The roots y are i and -i. Therefore, for y = i: |13i - 1|^2 = | -1 + 13i |^2 = 1 + 169 = 170. Similarly for y = -i: | -1 -13i |^2 = 1 + 169 = 170. Therefore, S = 170 + 170 = 340. Then the answer would be S / 2 = 170. However, in reality, the sum of 1/|r_i|^2 is 340. Wait, but according to our previous logic, the answer should be S / 2 = 340 / 2 = 170. But in reality, the sum of 1/|r_i|^2 is 340. Therefore, there is a contradiction here.

Wait, this suggests that my previous logic is wrong. What's happening here?

Wait, in this case, the original equation is x^2 + (13x - 1)^2 = 0. The roots are r and \overline{r}, so the sum we need is 1/(r \overline{r}) + 1/(\overline{r} r) = 2/(r \overline{r}). But according to the calculation, r \overline{r} = |r|^2 = 1/170, so the sum is 2/(1/170) = 340. However, according to the previous logic, S = sum |13y - 1|^2 = 340, and the answer is S / 2 = 170. But in reality, the answer is 340. Therefore, there is a mistake in my previous reasoning.

This suggests that my earlier conclusion that the answer is S / 2 is wrong. Therefore, I need to re-examine my logic.

Let me go back. In this simple case, the original equation is x^2 + (13x - 1)^2 = 0. The roots are x = [13 ± i]/170. Therefore, each root x corresponds to a y such that y = x/(13x - 1). Let me check.

For x = [13 + i]/170, compute y = x / (13x - 1). Let me compute 13x - 1: 13*(13 + i)/170 - 1 = (169 + 13i)/170 - 1 = (169 + 13i - 170)/170 = (-1 + 13i)/170. Therefore, y = [ (13 + i)/170 ] / [ (-1 + 13i)/170 ] = (13 + i)/(-1 + 13i). Multiply numerator and denominator by (-1 - 13i):

Numerator: (13 + i)(-1 - 13i) = -13 - 169i - i -13i^2 = -13 - 170i + 13 = -170i

Denominator: (-1)^2 - (13i)^2 = 1 - (-169) = 1 + 169 = 170

Therefore, y = (-170i)/170 = -i. Similarly, for the other root x = [13 - i]/170, y = i. Therefore, the roots y are i and -i, which satisfy y^2 = -1. Therefore, in this case, S = sum |13y - 1|^2 = |13i - 1|^2 + |13(-i) - 1|^2 = | -1 + 13i |^2 + | -1 - 13i |^2 = 1 + 169 + 1 + 169 = 340. Therefore, according to my previous logic, the answer should be S / 2 = 170, but in reality, the answer is 340. Therefore, my previous logic was wrong.

So where is the mistake?

Earlier, I thought that the sum over all roots of 1/|x|^2 is equal to S, and that the answer is S / 2. But in this case, the sum over all roots of 1/|x|^2 is 2 * (1/|r|^2) = 2 * 170 = 340, which is equal to S. Therefore, the answer is S, not S / 2. Therefore, my previous conclusion was wrong.

Wait, in this case, the sum over all roots of 1/|x|^2 is 340, which is equal to S. Therefore, the answer is 340, not S / 2. Therefore, my previous reasoning was incorrect.

So why was there a discrepancy? Let me re-examine.

Earlier, I thought that the sum over all roots of 1/|x|^2 is equal to S, and that the answer is half of that. But in reality, in this case, the answer is equal to S. Therefore, my mistake was in the assumption that the answer is half of S.

Let me re-examine the logic.

In the original problem, the sum we need is sum_{i=1}^5 1/(r_i \overline{r}_i). Each pair r_i and \overline{r}_i contributes one term, which is 1/(r_i \overline{r}_i) = 1/|r_i|^2. However, in the simple case, there are two roots, r and \overline{r}, and the sum is 1/(r \overline{r}) + 1/(\overline{r} r) = 2/(r \overline{r}) = 2/|r|^2. However, in this case, the sum over all roots of 1/|x|^2 is 2/|r|^2, which is equal to S. Therefore, the answer is equal to the sum over all roots of 1/|x|^2. Therefore, in the original problem, the answer is S, not S / 2.

Wait, but in the simple case, the answer is 340, which is equal to S. Therefore, my previous conclusion that the answer is S / 2 was wrong. Therefore, I need to re-examine the logic.

Let me go back.

In the original problem, the roots are r_1, \overline{r}_1, ..., r_5, \overline{r}_5. The sum we need is sum_{i=1}^5 1/(r_i \overline{r}_i). Each term 1/(r_i \overline{r}_i) is equal to 1/|r_i|^2. However, each pair r_i and \overline{r}_i corresponds to two roots, but the term 1/(r_i \overline{r}_i) is the same for both. Therefore, the sum over all roots of 1/|x|^2 is 2 * sum_{i=1}^5 1/|r_i|^2. Therefore, the answer is (sum over all roots of 1/|x|^2) / 2.

But in the simple case, the sum over all roots of 1/|x|^2 is 2/|r|^2, and the answer is 2/|r|^2, which would mean that the answer is equal to the sum over all roots of 1/|x|^2. Therefore, there is a contradiction.

Wait, in the simple case, the sum over all roots of 1/|x|^2 is 2/|r|^2, and the answer is also 2/|r|^2. Therefore, the answer is equal to the sum over all roots of 1/|x|^2. Therefore, my previous conclusion that the answer is S / 2 was wrong.

Wait, what is the relationship between the sum over all roots of 1/|x|^2 and the answer?

In the simple case, the answer is the sum over the pairs, which is 1/(r \overline{r}) + 1/(\overline{r} r) = 2/(r \overline{r}) = 2/|r|^2. But the sum over all roots of 1/|x|^2 is also 2/|r|^2. Therefore, the answer is equal to the sum over all roots of 1/|x|^2. Therefore, in the original problem, the answer is equal to S, not S / 2.

But why in the previous logic I thought that the answer is S / 2? Because I thought that the sum over all roots of 1/|x|^2 is twice the answer. But in reality, in the simple case, the answer is equal to the sum over all roots of 1/|x|^2.

Wait, let me clarify.

In the simple case, there are two roots: r and \overline{r}. The sum we need is 1/(r \overline{r}) + 1/(\overline{r} r) = 2/(r \overline{r}) = 2/|r|^2. However, the sum over all roots of 1/|x|^2 is also 2/|r|^2. Therefore, the answer is equal to the sum over all roots of 1/|x|^2.

Therefore, in the original problem, the answer is equal to the sum over all roots of 1/|x|^2, which is S = 1700. Therefore, the answer is 1700. But in the simple case, this would mean that the answer is 340, which matches.

But earlier, I thought that the answer was S / 2, but that was incorrect. Therefore, my mistake was in the previous logic.

So why was there confusion? Because I thought that the sum over the pairs is half of the total sum over all roots, but in reality, the sum over the pairs is equal to the total sum over all roots.

Wait, no. In the simple case, the sum over the pairs is 2/|r|^2, which is equal to the sum over all roots of 1/|x|^2. Therefore, the answer is equal to the sum over all roots of 1/|x|^2. Therefore, in the original problem, the answer is S = 1700.

But this contradicts the previous logic where I thought that the answer is S / 2. Where is the mistake?

Let me think again.

The problem says: the equation has 10 complex roots r1, \overline{r1}, ..., r5, \overline{r5}. Find the sum 1/(r1 \overline{r1}) + ... + 1/(r5 \overline{r5}).

Each term 1/(r_i \overline{r_i}) is equal to 1/|r_i|^2. Therefore, the sum is sum_{i=1}^5 1/|r_i|^2.

However, the sum over all roots of 1/|x|^2 is sum_{i=1}^{10} 1/|x_i|^2 = sum_{i=1}^5 [1/|r_i|^2 + 1/|\overline{r_i}|^2] = sum_{i=1}^5 2/|r_i|^2. Therefore, the sum over all roots is twice the desired sum. Therefore, the desired sum is (sum over all roots of 1/|x|^2) / 2.

But in the simple case, the sum over all roots of 1/|x|^2 is 2/|r|^2, and the desired sum is also 2/|r|^2. Therefore, this suggests that (sum over all roots of 1/|x|^2) / 2 = 2/|r|^2 / 2 = 1/|r|^2, which contradicts.

Wait, this is confusing. Let me clarify with the simple case.

In the simple case, the roots are r and \overline{r}. The desired sum is 1/(r \overline{r}) + 1/(\overline{r} r) = 2/(r \overline{r}) = 2/|r|^2.

The sum over all roots of 1/|x|^2 is 1/|r|^2 + 1/|\overline{r}|^2 = 2/|r|^2.

Therefore, the desired sum is equal to the sum over all roots of 1/|x|^2.

But according to the previous logic, the desired sum should be (sum over all roots of 1/|x|^2) / 2. But in reality, it is equal to the sum over all roots of 1/|x|^2.

This suggests that my previous logic was wrong. Where is the mistake?

Ah! The problem says that the roots are r1, \overline{r1}, ..., r5, \overline{r5}. Therefore, the sum we need is sum_{i=1}^5 1/(r_i \overline{r_i}) = sum_{i=1}^5 1/|r_i|^2. However, the sum over all roots of 1/|x|^2 is sum_{i=1}^{10} 1/|x_i|^2 = sum_{i=1}^5 [1/|r_i|^2 + 1/|r_i|^2] = 2 * sum_{i=1}^5 1/|r_i|^2. Therefore, the desired sum is (sum over all roots of 1/|x|^2) / 2.

But in the simple case, this would mean that the desired sum is (2/|r|^2) / 2 = 1/|r|^2, but in reality, the desired sum is 2/|r|^2. Therefore, there is a contradiction.

Wait, what is the desired sum in the simple case? The problem says: "the equation has 10 complex roots r1, \overline{r1}, ..., r5, \overline{r5}, where the bar denotes complex conjugation. Find the value of ...". In the simple case, the equation is degree 2, so it has 2 roots, which are r1 and \overline{r1}. Therefore, the sum is 1/(r1 \overline{r1}) = 1/|r1|^2. But in our calculation, the sum over all roots of 1/|x|^2 is 2/|r1|^2, and the desired sum is 1/|r1|^2, which is half of the total sum. Therefore, in this case, the desired sum is (sum over all roots of 1/|x|^2) / 2.

But earlier, I thought that the desired sum was 2/|r|^2, but that was incorrect. Let me re-examine the simple case.

In the simple case, the equation is x^2 + (13x - 1)^2 = 0, which has two roots: r and \overline{r}. The problem would say that the equation has 2 complex roots r1, \overline{r1}, and the sum is 1/(r1 \overline{r1}). Therefore, the desired sum is 1/|r1|^2. However, in our calculation, the sum over all roots of 1/|x|^2 is 2/|r1|^2, so the desired sum is half of that.

But earlier, I thought that the desired sum was 2/|r|^2, but that was a mistake. The correct desired sum is 1/|r|^2, which is half of the total sum over all roots.

Therefore, in the original problem, the desired sum is (sum over all roots of 1/|x|^2) / 2 = S / 2 = 1700 / 2 = 850.

But why did I get confused earlier? Because I incorrectly calculated the desired sum as 2/|r|^2, but actually, it's 1/|r|^2.

Let me verify with the simple case.

In the simple case, the roots are r and \overline{r}. The problem says to find 1/(r \overline{r}) + ... but since there is only one pair, it's just 1/(r \overline{r}) = 1/|r|^2. The sum over all roots of 1/|x|^2 is 1/|r|^2 + 1/|\overline{r}|^2 = 2/|r|^2. Therefore, the desired sum is half of that, which is 1/|r|^2.

Therefore, in the original problem, the desired sum is S / 2 = 1700 / 2 = 850.

But in the simple case, S = 340, and the desired sum is 170, which is S / 2. But earlier, I thought that the desired sum was 340, but that was incorrect.

Wait, let me recalculate the simple case.

In the simple case, the equation is x^2 + (13x - 1)^2 = 0. The roots are x = [13 ± i]/170. Therefore, |x|^2 = (13^2 + 1^2)/170^2 = 170 / 28900 = 1/170. Therefore, 1/|x|^2 = 170. Therefore, the sum over all roots of 1/|x|^2 is 170 + 170 = 340. Therefore, the desired sum is 340 / 2 = 170.

But earlier, I thought that the desired sum was 2/|r|^2, but that was incorrect. The correct desired sum is 170, which is S / 2.

Therefore, my initial conclusion was correct, and the confusion was due to miscalculating the desired sum in the simple case.

Therefore, returning to the original problem, the answer is S / 2 = 1700 / 2 = 850.

Therefore, the correct answer is 850.

But to confirm, let me check with the simple case. If the answer is 170, which is S / 2 = 340 / 2 = 170, and in reality, 1/|x|^2 for each root is 170, so the sum over all roots is 340, and the desired sum is 170.

Therefore, the logic holds.

Therefore, the answer is 850.

**Final Answer**
\boxed{850}

To solve the equation

$$
x^{10} + (13x - 1)^{10} = 0,
$$

we are given that it has 10 complex roots, which come in conjugate pairs: $ r_1, \overline{r}_1, r_2, \overline{r}_2, \ldots, r_5, \overline{r}_5 $. The goal is to compute the sum:

$$
\frac{1}{r_1\overline{r}_1} + \frac{1}{r_2\overline{r}_2} + \cdots + \frac{1}{r_5\overline{r}_5}.
$$

---

### Step 1: Understanding the Expression

For any complex number $ z $, the product $ z\overline{z} = |z|^2 $, the square of its modulus. Therefore, the expression becomes:

$$
\sum_{i=1}^5 \frac{1}{r_i \overline{r}_i} = \sum_{i=1}^5 \frac{1}{|r_i|^2}.
$$

So, we are to compute the sum of the reciprocals of the squared moduli of the roots.

---

### Step 2: Rewriting the Equation

We can rewrite the original equation as:

$$
x^{10} = - (13x - 1)^{10}.
$$

Taking both sides to the power of $ \frac{1}{10} $, we get:

$$
\left( \frac{x}{13x - 1} \right)^{10} = -1.
$$

Let $ y = \frac{x}{13x - 1} $. Then $ y^{10} = -1 $, so $ y $ is a 10th root of $-1$. The 10th roots of $-1$ are the complex numbers $ y = e^{i(\pi + 2\pi k)/10} $ for $ k = 0, 1, \ldots, 9 $, which are all on the unit circle.

Solving for $ x $ in terms of $ y $, we get:

$$
x = \frac{y}{13y - 1}.
$$

So each root $ x $ corresponds to a root $ y $ of $ y^{10} = -1 $.

---

### Step 3: Expressing the Desired Sum

We want to compute:

$$
\sum_{i=1}^5 \frac{1}{|r_i|^2} = \sum_{i=1}^5 \frac{1}{|x_i|^2}.
$$

From the expression $ x = \frac{y}{13y - 1} $, we can compute:

$$
|x|^2 = \left| \frac{y}{13y - 1} \right|^2 = \frac{|y|^2}{|13y - 1|^2}.
$$

Since $ |y| = 1 $ (because $ y $ lies on the unit circle), this simplifies to:

$$
|x|^2 = \frac{1}{|13y - 1|^2} \quad \Rightarrow \quad \frac{1}{|x|^2} = |13y - 1|^2.
$$

Thus, the sum becomes:

$$
\sum_{i=1}^5 \frac{1}{|r_i|^2} = \sum_{i=1}^5 |13y_i - 1|^2,
$$

where $ y_i $ are the 10th roots of $-1$, and we are summing over 5 distinct pairs of conjugate roots.

However, since the roots $ y $ come in conjugate pairs, and $ |13y - 1|^2 = |13\overline{y} - 1|^2 $, each pair contributes the same value. Therefore, the total sum over all 10 roots is:

$$
\sum_{y^{10} = -1} |13y - 1|^2 = 2 \sum_{i=1}^5 |13y_i - 1|^2.
$$

So, the desired sum is:

$$
\sum_{i=1}^5 \frac{1}{|r_i|^2} = \frac{1}{2} \sum_{y^{10} = -1} |13y - 1|^2.
$$

---

### Step 4: Computing the Sum

We now compute:

$$
S = \sum_{y^{10} = -1} |13y - 1|^2.
$$

Using the identity $ |a - b|^2 = (a - b)(\overline{a} - \overline{b}) $, and noting that $ |y| = 1 $, we have:

$$
|13y - 1|^2 = (13y - 1)(13\overline{y} - 1) = 169 - 13y - 13\overline{y} + 1 = 170 - 13(y + \overline{y}).
$$

Since $ y $ are the 10th roots of $-1$, and $ y^{10} = -1 $, the sum of all such $ y $ is zero (by Vieta's formula, since the polynomial $ y^{10} + 1 = 0 $ has zero coefficient for $ y^9 $). Similarly, the sum of $ \overline{y} $ is also zero. Therefore:

$$
\sum_{y^{10} = -1} |13y - 1|^2 = \sum_{y^{10} = -1} (170 - 13(y + \overline{y})) = 10 \cdot 170 - 13 \cdot 0 = 1700.
$$

Thus, the desired sum is:

$$
\frac{1}{2} \cdot 1700 = \boxed{850}.
$$

---

### Final Answer

$$
\boxed{850}
$$

Solution:
Let $p(x) = x^{10} + (13x - 1)^{10}.$  If $r$ is a root of $p(x),$ then $r^{10} + (13x - 1)^{10} = 0.$  Then $(13r - 1)^{10} = -r^{10},$ so
\[-1 = \left( \frac{13r - 1}{r} \right)^{10} = \left( \frac{1}{r} - 13 \right)^{10}.\]Then $\frac{1}{r} - 13$ has magnitude 1, so
\[\left( \frac{1}{r} - 13 \right) \left( \frac{1}{\overline{r}} - 13 \right) = 1,\]so
\[\left( \frac{1}{r_1} - 13 \right) \left( \frac{1}{\overline{r}_1} - 13 \right) + \dots + \left( \frac{1}{r_5} - 13 \right) \left( \frac{1}{\overline{r}_5} - 13 \right) = 5.\]Expanding, we get
\[\frac{1}{r_1 \overline{r}_1} + \dots + \frac{1}{r_5 \overline{r}_5} - 13 \left( \frac{1}{r_1} + \frac{1}{\overline{r}_1} + \dots + \frac{1}{r_5} + \frac{1}{\overline{r}_5} \right) + 5 \cdot 169 = 5.\]We see that $\frac{1}{r_1},$ $\frac{1}{\overline{r}_1},$ $\dots,$ $\frac{1}{r_5},$ $\frac{1}{\overline{r}_5}$ are the solutions to
\[\left( \frac{1}{x} \right)^{10} + \left( \frac{13}{x} - 1 \right)^{10} = 0,\]or $1 + (13 - x)^{10} = 0.$  The first few terms in the expansion as
\[x^{10} - 130x^9 + \dotsb = 0,\]so by Vieta's formulas,
\[\frac{1}{r_1} + \frac{1}{\overline{r}_1} + \dots + \frac{1}{r_5} + \frac{1}{\overline{r}_5} = 130.\]Hence,
\[\frac{1}{r_1 \overline{r}_1} + \dots + \frac{1}{r_5 \overline{r}_5} = 13 \cdot 130 - 5 \cdot 169 + 5 = \boxed{850}.\]