MathLabs

Problem 5

Determine all possible values of S=aa+b+d+ba+b+c+cb+c+d+da+c+dS=\dfrac{a}{a+b+d}+\dfrac{b}{a+b+c}+\dfrac{c}{b+c+d}+\dfrac{d}{a+c+d}, where a,b,c,da,b,c,d are arbitrary positive real numbers.
Step 2 of 4: Choose a largest variable for the upper bound
Previous stepNext step
a=max⁡{a,b,c,d}a=\max\{a,b,c,d\}
Detailed analysis

Assume a is a largest variable. The first fraction is less than 1. In each of the other three fractions, replacing its denominator by b+c+d can only increase it, because a is at least each omitted variable. Thus the sum of the last three fractions is at most 1, and S<2.