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jmc

algebra intermediate

Problem

Let

r,

s,

and

t

be the roots of the equation

x^{3} - 20 x^{2} + 18 x - 7 = 0.

Find the value of

\frac{r}{\frac{1}{r} + s t} + \frac{s}{\frac{1}{s} + t r} + \frac{t}{\frac{1}{t} + r s} .

Solution

Note that

\frac{r}{\frac{1}{r} + s t} = \frac{r ^{2}}{1 + r s t} = \frac{r ^{2}}{1 + 7} = \frac{r ^{2}}{8},

since

r s t = 7

by Vieta's formulas. By similar computations, we get

\frac{r}{\frac{1}{r} + s t} + \frac{s}{\frac{1}{s} + t r} + \frac{t}{\frac{1}{t} + r s} = \frac{r ^{2} + s ^{2} + t ^{2}}{8},

which equals

\frac{( r + s + t ) ^{2} - 2 ( r s + s t + t r )}{8} = \frac{2 0 ^{2} - 2 \cdot 18}{8} = \frac{91}{2} .

Final answer

\frac{91}{2}