$#EA$

`a^2/(b+c) + b^2/(c+a) + c^2/(a+b) ≥ 1/2`

Áp dụng bđt Cauchy, ta có :

`a^2/(b+c) + (b+c)/4 ≥ 2 sqrt{a^2/(b+c) . (b+c)/4}`

`→` `a^2/(b+c) + (b+c)/4 ≥ 2 sqrt{a^2/4}`

`→` `a^2/(b+c) + (b+c)/4 ≥ a`     `(1)` 

`b^2/(c+a) + (c+a)/4 ≥ 2 sqrt{b^2/(c+a) . (c+a)/4}`

`→` `b^2/(c+a) + (c+a)/4 ≥ 2 sqrt{b^2/4}`

`→` `b^2/(c+a) + (c+a)/4 ≥ b`     `(2)` 

`c^2/(a+b) + (a+b)/4 ≥ 2 sqrt{c^2/(a+b) . (a+b)/4}`

`→` `c^2/(a+b) + (a+b)/4 ≥ 2 sqrt{c^2/4}`

`→` `c^2/(a+b) + (a+b)/4 ≥ c`     `(3)` 

Từ `(1) + (2) + (3)` :

`→`  `(a^2/(b+c) + (b+c)/4) + (b^2/(c+a) + (c+a)/4) + (c^2/(a+b) + (a+b)/4) ≥ a + b + c`

`→` `a^2/(b+c) + b^2/(c+a) + c^2/(a+b) ≥ a + b + c - ( (b+c)/4 + (c+a)/4 + (a+b)/4 )`

`→` `a^2/(b+c) + b^2/(c+a) + c^2/(a+b) ≥ (a+b+c)/2`

`→` `a^2/(b+c) + b^2/(c+a) + c^2/(a+b) ≥ 1/2`

Dấu " `=` " xảy ra khi `a = b = c = 1/3` .