Prove that `2/(b+c)+2/(c+a)+2/(a+b)lt1/a+1/b+1/c `,where `a ,b ,c>0.`

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

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 .

Prove that: |1a a^2-b c1bb^2-c a1cc^2-a b|=0

"If " a^(2), b^(2), c^(2)" are in A.P., prove that "(1)/(b+c),(1)/(c+a),(1)/(a+b) " are also in A.P."

Prove that : |{:(a+b,b+c,c+a),(c,a,b),(1,1,1):}|

if (b-c)^(2) , (c-a)^(2) ,(a-b)^(2) are in AP, prove that 1/((b-c)) ,1/(( c-a)) ,1 /((a-b)) are in AP.

If (b-c)^(2),(c-a)^(2),(a-b)^(2) are in A.P.then prove that (1)/(b-c),(1)/(c-a),(1)/(a-b) are also in A.P.

If a(1/b +1/c) , b (1/c +1/a),c(1/a +1/b) are in AP , prove tht a^(2)(b+c) ,b^(2) (c+a) , c^(2) (a+b) are in AP.

If a^(2),b^(2),c^(2) are in AP then prove that (1)/(a+b),(1)/(c+b),(1)/(a+c) are also in AP.