Problem 5

Determine all possible values of

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

, where

a,b,c,d

are arbitrary positive real numbers.

Step 4 of 4: Approach the lower endpoint and use continuity

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a=1,\quad b=t,\quad c=t^2,\quad d=t^3\to0^+

Detailed analysis

For $a=1$ , $b=t$ , $c=t^2$ , and $d=t^3$ , the first term tends to $1$ while each of the other three tends to $0$ , so $S$ approaches $1$ from above. The domain of positive quadruples is connected and $S$ is continuous; together with the strict bounds and both endpoint limits, its image is exactly the open interval $(1,2)$ .