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Part of 2017 Caucasus Mathematical Olympiad

Problems(2)

Source: II Caucasus Mathematical Olympiad

Basil needs to solve an exercise on summing two fractions

\dfrac{a}{b}

\dfrac{c}{d}

a

b

c

d

are some non-zero real numbers. But instead of summing he performed multiplication (correctly). It appears that Basil's answer coincides with the correct answer to given exercise. Find the value of

\dfrac{b}{a} + \dfrac{d}{c}

another easy problem

Source: II Caucasus Mathematical Olympiad

A

B

lie on two branches of hyperbola given by equation

y=\frac1x

A_x

A_y

be projections of

A

onto coordinate axis, similarly, let

B_x

B_y

be projections of

B

onto coordinate axis. Prove that triangles

AB_xB_y

BA_xA_y

have equal areas.