Question 31
Let $a, b, c, d$ be four positive real numbers. If
$$
a+b+\frac{1}{ab}=c+d+\frac{1}{cd} \quad \text{and} \quad \frac{1}{a}+\frac{1}{b}+ab=\frac{1}{c}+\frac{1}{d}+cd
$$
hold, prove that at least two of the values $a, b, c, d$ are equal.
$$
a+b+\frac{1}{ab}=c+d+\frac{1}{cd} \quad \text{and} \quad \frac{1}{a}+\frac{1}{b}+ab=\frac{1}{c}+\frac{1}{d}+cd
$$
hold, prove that at least two of the values $a, b, c, d$ are equal.