If `a ,b ,c` are real numbers such that `0 < a < 1,0 < b < 1,0 < c < 1,a+b+c=2,` then prove that `a/(1-a)b/(1-b)c/(1-c)geq8`

If a , b , c , are real number such that a c!=0, then show that at least one of the equations a x^2+b x+c=0 and -a x^2+b x+c=0 has real roots.

If a ,\ b ,\ c are real numbers such that |(b+c,c+a ,a+b),( c+a,a+b,b+c),(a+b,b+c,c+a)|=0 , then show that either a+b+c=0 or, a=b=c .

If a,b,c are real numbers such that 3(a^(2)+b^(2)+c^(2)+1)=2(a+b+c+ab+bc+ca) , than a,b,c are in

Let a ,b ,and c be real numbers such that 4a+2b+c=0 and a b > 0. Then the equation a x^2+b x+c = 0

If a,b,c are positive real numbers such that a + b +c=18, find the maximum value of a^2b^3c^4

If a,b,c are real distinct numbers such that a ^(3) +b ^(3) +c ^(3)= 3abc, then the quadratic equation ax ^(2) +bx +c =0 has (a) Real roots (b) At least one negative root (c) Both roots are negative (d) Non real roots

If a,b,c are non-zero real number such that |(bc,ca,ab),(ca,ab,bc),(ab,bc,ca)|=0, then

If a,b,c are non-zero real number such that |(bc,ca,ab),(ca,ab,bc),(ab,bc,ca)|=0, then

Let a, b and c be real numbers such that 4a + 2b + c = 0 and ab gt 0. Then the equation ax^(2) + bx + c = 0 has

Let a ,b and c be real numbers such that a+2b+c=4 . Find the maximum value of (a b+b c+c a)dot