SOLUTION: What are 3 unequal positive rational numbers a, b, & c for which a+b+c=1/(a+b+c)

a+b+c = 

Multiply both sides by (a+b+c)

(a+b+c)² = 1

Take positive square roots of both sides:

a+b+c = 1

So we just need three fractions that have sum = 1

To do that take any 3 integers, say 1, 4 and 5

Put their sum over itself and that will equal 1

 = 1 

Add the terms in the denominator but leave the numerator
terms added:

 = 1

Write as the sum of three fractions:

 +  +  = 1

Reduce any that will reduce:

 +  +  = 1

So  a=, b=, c=

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There is no end to the number of different possibilities following that recipe:

 =  =  +  +  =  +  +  = 1

 =  =  +  +  =  +  +  = 1

 =  =  +  +  =  +  +  = 1

Edwin