Prove that `a^4+b^4+c^4> a b c(a+b+c),w h e r ea ,b ,c > 0.`

Given a matrix A=[a b c b c a c a b],w h e r ea ,b ,c are real positive numbers a b c=1a n dA^T A=I , then find the value of a^3+b^3+c^3dot

Given a matrix A=[a b c b c a c a b],w h e r ea ,b ,c are real positive numbers a b c=1a n dA^T A=I , then find the value of a^3+b^3+c^3dot

In Delta A B C , prove that t a n A+t a n B+t a n Cgeq3sqrt(3),w h e r eA ,B ,C are acute angles.

If |1 1 1a b c a^3b^2c^3|=(a-b)(b-c)(c-a)(a+b+c),w h e r ea ,b ,c are different, then the determinant |1 1 1(x-a)^2(x-b)^2(x-c)^2(x-b)(x-c)(x-c)(x-a)(x-a)(x-b)| vanishes when a. a+b+c=0 b. x=1/3(a+b+c) c. x=1/2(a+b+c) d. x=a+b+c

If f(x)=a x^2+b x+a^2+b^2+c^2-a b-b c-c a ,w h e r ea ,b ,c are distinct reals, has imaginary roots then (a) (a-b)^2+(b-c)^2+(c-a)^2>0 (b) f(0)>0 (c) f(-1)>0 (d) f(0)<0

If the equation of plane containing the line x-y+z=0a n dx+y+z-2=0 and which is farthest from origin is a x+b y+c z=d ,w h e r ea , b ,c ,d in N , then find the minimum value of (a+b+c+d)dot

Suppose A, B, C are defined as A=a^2b+a b^2-a^2c-a c^2 , B=b^2c+b c^2-a^2b-a b^2,and C=a^2c +'a c^2-b^2' c-b c^2, w h e r ea > b > c >0 and the equation A x^2+B x+C=0 has equal roots, then a ,b ,c are in AdotPdot b. GdotPdot c. HdotPdot d. AdotGdotPdot

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

Use the factor theorem to find the value of k for which (a+2b),w h e r ea ,b!=0 is a factor of a^4+32 b^4+a ^3 b(k+3)dot

Find the number of polynomials of the form x^3+a x^2+b x+c that are divisible by x^2+1,w h e r ea , b ,c in {1,2,3,9,10}dot