If `a , b ,c` are three distinct positive real numbers in G.P., then prove that `c^2+2a b >3a cdot`

If a ,b ,c are three distinct real numbers in G.P. and a+b+c=x b , then prove that either x >3.

If a ,b ,c are three distinct real numbers in G.P. and a+b+c=x b , then prove that either x >3.

If a ,b ,c are three distinct positive real numbers, the number of real and distinct roots of a |x^2|+2b|x|-c=0 is a. 0 b. 4 c. 2 d. none of these

If a,b and c are distinct positive real numbers in A.P, then the roots of the equation ax^(2)+2bx+c=0 are

If the parabolas y^2=4a x and y^2=4c(x-b) have a common normal other than the x-axis (a , b , c being distinct positive real numbers), then prove that b/(a-c)> 2.

If a ,b , a n dc 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 a, b, c are distinct positive real numbers in G.P and log_ca, log_bc, log_ab are in A.P, then find the common difference of this A.P

If a , b , c , are positive real numbers, then prove that (2004, 4M) {(1+a)(1+b)(1+c)}^7>7^7a^4b^4c^4

If a, b and c are distinct positive numbers, then the expression (a + b - c)(b+ c- a)(c+ a -b)- abc is:

Let a ,b ,c ,d be four distinct real numbers in A.P. Then half of the smallest positive valueof k satisfying 2(a-b)+k(b-c)^2+(c-a)^3=2(a-d)+(b-d)^2+(c-d)^3 is __________.