Prove that `(b^2+c^2)/(b+c)+(c^2+a^2)/(c+a)+(a^2+b^2)/(a+b)> a+b+c`

Prove that (b^2+c^2)/(b+c)+(c^2+a^2)/(c+a)+(a^2+b^2)/(a+b)gt a+b+c , where a,b,cgt 0 .

If (b^2+c^2-a^2)/(2b c),(c^2+a^2-b^2)/(2c a),(a^2+b^2-c^2)/(2a b) are in A.P. and a+b+c=0 then prove that a(b+c-a),b(c+a-b),c(a+b-c) are in A.P.

Prove that (2)/(b+c)+(2)/(c+a)+(2)/(a+b) 0.

Prove that : |{:(b^(2)c^(2),bc, b+c),(c^(2)a^(2),ca, c+a),(a^(2)b^(2),ab, a+b):}|=0

Prove that |(a^2,b^2+c^2,bc),(b^2,c^2+a^2,ca),(c^2,a^2+b^2,ab)|=-(a-b)(b-c)(c-a)(a+b+c)(a^2+b^2+c^2)

Prove: |(1,b+c ,b^2+c^2),( 1,c+a ,c^2+a^2),( 1,a+b ,a^2+b^2)|=(a-b)(b-c)(c-a)

Prove that a(b^(2) + c^(2)) cos A + b(c^(2) + a^(2)) cos B + c(a^(2) + b^(2)) cos C = 3abc

prove that b^(2)c^(2)+c^(2)a^(2)+a^(2)b^(2)>=abc(a+b+c)

Prove that b^(2)c^(2)+c^(2)a^(2)+a^(2)+b^(2)>abc xx(a+b+c)(a,b,c>0)