Prove that
`|(1+a_1,1,1),(1,1+a_2,1),(1,1,1+a_3)|=a_1a_2a_3(1+1/a_1+1/a_2+1/a_3)` provided `a_1 ≠ 0, a_2 ≠ 0 and a_3 ≠ 0`.

Evaluate: /_\ |[1+a_1, a_2, a_3],[a_1, 1+a_2, a_3],[a_1, a_2, 1+a_3]|

Evaluate: /_\ |[1+a_1, a_2, a_3],[a_1, 1+a_2, a_3],[a_1, a_2, 1+a_3]|

Evaluate: /_\ |[1+a_1, a_2, a_3],[a_1, 1+a_2, a_3],[a_1, a_2, 1+a_3]|

prove that |(1/(a-a_1)^2,1/(a-a_1),1/a_1),(1/(a-a_2)^2,1/(a-a_2),1/a_2),(1/(a-a_3)^2,1/(a-a_3),1/a_3)|=(-a^2(a_1-a_2)(a_2-a_3)(a_3-a_1))/(a_1a_2a_3(a-a_1)^2(a-a_2)^2(a-a_3)^2) .

Prove that ((a_1)/(a_2)+(a_3)/(a_4)+(a_5)/(a_6)) ((a_2)/(a_1)+(a_4)/(a_3)+(a_6)/(a_5)) geq 9

If (a_2 a_3)/(a_1 a_4) = (a_2 + a_3)/(a_1 + a_4) = 3 ((a_2 - a_3)/(a_1 - a_4)) then a_1, a_2, a_3, a_4 are in

If (a_2a_3)/(a_1a_4) = (a_2+a_3)/(a_1+a_4)=3 ((a_2 -a_3)/(a_1-a_4)) then a_1,a_2, a_3 , a_4 are in :

If (a_2a_3)/(a_1a_4) = (a_2+a_3)/(a_1+a_4)=3 ((a_2 -a_3)/(a_1-a_4)) then a_1,a_2, a_3 , a_4 are in :