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 3 of 4: Approach the upper endpoint
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a=c=1,b=d=t→0+a=c=1,\quad b=d=t\to0^+
Detailed analysis

For a=c=1 and b=d=t, the first and third terms tend to 1 while the other two tend to 0. Hence S can be made arbitrarily close to 2 from below.