If `a+b+c=1,` then prove that `8/(27a b c)>{1/a-1}{1/b-1}{1/c-1}> 8.`

If a,b, and c are distinct positive real numbers such that a+b+c=1, then prove tha ((1+a)(1+b)(1+c))/((1-a)(1-b)(1-c))>8

If b is the harmonic mean between a and c, then prove that (1)/(b - a) + (1)/(b - c) = (1)/(b) + (1)/(c)

If b is the harmonic mean between a and c, then prove that (1)/(b-a)+(1)/(b-c)=(1)/(a)+(1)/(c)

If a,b,c are real numbers such that 0 lt a lt 1, 0 lt b gt 1, 0 lt c lt 1, a + b + c = 2 , then prove that (a)/(1 - a) (b)/(1 - b) (c )/(1 - c) ge 8 .

If a > b > c >0 , prove that cot^(-1)((a b+1)/(a-b))+cot^(-1)((b c+1)/(b-c))+cot^(-1)((c a+1)/(c-a))=pi

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

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

If a ,b and c are all non-zero and |[1+a,1, 1],[ 1,1+b,1],[1,1,1+c]|=0, then prove that 1/a+1/b+1/c+1=0

If 1/(a+b)+1/(b+c)=1/b , prove that a,b,c are in G.P.

IF (1)/(a),(1)/(b),(1)/(c) are in AP then prove that (b+c)/(a),(a+c)/(b),(a+b)/(c) are in AP