Problem 2

If a + b + c = 7 and (1)/(a + b) + (1)/(b + c) + (1)/(c + a) = (7)/(10), what is (a)/(b + c) + (b)/(c + a) + (c)/(a + b)?

Solution.

We have

(a + b + c)⋅⎛⎝(1)/(a + b) + (1)/(b + c) + (1)/(c + a)⎞⎠ = 7⋅(7)/(10) = (49)/(10).

We also have:

(a + b + c)⋅⎛⎝(1)/(a + b) + (1)/(b + c) + (1)/(c + a)⎞⎠

= (a + b + c)/(a + b) + (a + b + c)/(b + c) + (a + b + c)/(c + a)

= 3 + (a)/(b + c) + (b)/(c + a) + (c)/(a + b).

This gives us

3 + (a)/(b + c) + (b)/(c + a) + (c)/(a + b) = (49)/(10),

then

(a)/(b + c) + (b)/(c + a) + (c)/(a + b) = (19)/(10).