Prove that `b^2c^2+c^2a^2+a^2+b^2> a b cxx(a+b+c)(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)+c^(2)a^(2)+a^(2)b^(2)>=abc(a+b+c)

Prove that 2b^(2)c^(2) +2c^(2)a^(2) +2a^(2)b^(2) -a^(4)-b^(4)-c^(4)= (a+b+c) (b+c-a) (c+a-b) (a+b-c)

Prove: |(a^2,b c, a c+c^2),(a^2+a b,b^2,a c ),(a b,b^2+b c,c^2)|=4a^2b^2c^2

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

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

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 |[a,b,c] , [a^2,b^2,c^2] , [a^3,b^3,c^3]|= abc(a-b)(b-c)(c-a)

prove that [ [a-b-c , 2a , 2a ] , [2b , b-c-a , 2b ] ,[2c ,2c,c-a-b]]= (a+b+c)^3

Prove: |(-bc, b^2+b c,c^2+bc), (a^2+a c,-a c,c^2+a c),( a^2+a b,b^2+a b,-a b)|=(a b+b c+c a)^3