Find-the-minimum-value-of-a-b-c-b-c-a-c-a-b-for-all-positive-real-numbers-

Question Number 198903 by necx122 last updated on 25/Oct/23

$$Findtheminimumvalueof$$$$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}forallpositivereal$$$$numbers$$

Commented by necx122 last updated on 25/Oct/23

please I need help with this

Answered by AST last updated on 25/Oct/23

$$=\frac{a^2}{ab+ac}+\frac{b^2}{bc+ab}+\frac{c^2}{ac+bc}$$$$⩾\frac{{(a+b+c)}^2=a^2+b^2+c^2+2(ab+bc+ca)}{2(ab+bc+ca)}⩾\frac{3(ab+bc+ca)}{2(ab+bc+ca)}$$$$=\frac{3}{2}[Equalitywhena=b=c]$$

Commented by necx122 last updated on 25/Oct/23

wow!!! This is a great solution sir. Please, I'm quite new to this manner of questions. Sir AST, can you please suggested sources where I can learn this properly. Thank you.

Commented by AST last updated on 25/Oct/23

$$ThisisjustanapplicationofCauchy-Schwarz$$$$\left({Engel}^{\prime }sForm\right).$$

Commented by necx122 last updated on 25/Oct/23

wow! This sounds like a sort of advanced level maths but how come I'm seeing such a question in junior mathematics olympiad? Could it mean that students in the junior school are really expected to know this? It's indeed overwhelming.

Commented by AST last updated on 25/Oct/23

$$Notreallyadvanced,theyarestandardtechniques$$$$in“{olympiad}^{″}competitions.$$

Commented by necx122 last updated on 25/Oct/23

hmmm... This is indeed a lot. Thank you for this information sir. I'm grateful.

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