If `=|a b c b^2cc^2b a b cc^2a c a^2a b c a^2bb^2a|=0,(a ,b ,c in R` and are all different and nonzero), then prove that `a+b+c=0.`

If =|[a b c, b^2c,c^2b],[ a b c,c^2a, c a^2],[a b c, a^2b,b^2a]|=0,(a ,b ,c in R) and are all different and nonzero), then prove that a+b+c=0.

" if " Delta = |{:(abc,,b^(2)c,,c^(2)b),(abc ,,c^(2)a,,ca^(2)),( abc,,a^(2)b,,b^(2)a):}| =0 , (a, b, c in R " and are all " different and non- zero ) the prove that a+b+c=0

Show that: |b^2+c^2a b a c b a c^2+a^2b cc a c b a^2+b^2|=4a^2\ b^2\ c^2 .

Prove that b^2c^2+c^2a^2+a^2b^2> a b cxx(a+b+c)(a ,b ,c >0) .

Without expanding show that |b^2c^2 bc b+cc^2a^2 ca c+a a^2b^2a b a+b|=0 .

If a^2+b^2+c^2=20 and a+b+c=0 , find a b+b c+c a

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

If a ,b ,c are nonzero real numbers and a z^2+b z+c+i=0 has purely imaginary roots, then prove that a=b^2c

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