HMMT February

Algebra Difficulty 5.0 AIME, harder Find the answer

Real numbers $a, b, c$ satisfy the equations $a+b+c=26,1 / a+1 / b+1 / c=28$ . Find the value of $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+\frac{a}{c}+\frac{c}{b}+\frac{b}{a}$

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Multiplying the two given equations gives $\frac{a}{a}+\frac{a}{b}+\frac{a}{c}+\frac{b}{a}+\frac{b}{b}+\frac{b}{c}+\frac{c}{a}+\frac{c}{b}+\frac{c}{c}=26 \cdot 28=728$ and subtracting 3 from both sides gives the answer, 725.

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