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Algebra Difficulty 4.6 AIME Find the answer Slovenia

Non-zero real numbers $a$ and $b$ satisfy $\frac{a}{b+1} + \frac{b}{a+1} = 1$ . Which of the statements about the expression $\frac{a}{b} + \frac{b}{a} - \frac{1}{ab}$ is correct?

Pick one

Solution

Remove the fractions and simplify the equation to get $a^2 + b^2 = ab + 1$ . Change the fractions in the given expression to the common denominator to obtain $\frac{a^2 + b^2 - 1}{ab}$ . The equality now implies that the value of the expression is $\frac{ab}{ab} = 1$ . The correct answer is (C).

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.