Prove that `(a^8+b^8+c^8)/(b+c)gt (1)/(a)+(1)/(b) +(1)/(c)` .

Prove that (a^8+b^8+c^8)/(a^3b^3c^3)>1/a+1/b+1/c

Prove that (1)/(a) + (1)/(b) + (1)/(c ) ge (1)/(sqrt((bc))) + (1)/(sqrt((ca))) + (1)/(sqrt((ab))) , where a,b,c gt 0

Prove that (cos C + cos A)/(c + a) + (cos B)/(b) = (1)/(b)

Prove that |{:(1,,ab,,(1)/(a)+(1)/(b)),(1 ,,bc,,(1)/(b)+(1)/(c)),( 1,, ca,,(1)/(c)+(1)/(a)):}|=0

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

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.

Prove that |(1/a^(2),bc,b+c),(1/b^(2),ca,c+a),(1/c^(2), ab, a+b)|=0

Prove that |(1+a,1,1),(1,1+b,1),(1,1,1+c)|=abc(1+(1)/(a)+(1)/(b)+(1)/(c)).

In a triangle ABC, prove that (a^(2)+b^(2))/(a^(2)+c^(2))=(1+cos (A-B) cos C)/(1+cos (A-C) cos B)