If `a^b = b^a` then prove that, `(a/b)^(a/b) = a^(b/a-1)`

Prove that: (a-b)^3=a^3-b^3-3a b(a-b)

If b is the harmonic mean between a and c, then prove that (1)/(b - a) + (1)/(b - c) = (1)/(a) + (1)/(c)

If a, b, c are positive real numbers such that a + b + c = 1 , then prove that a/(b + c)+b/(c+a) + c/(a+b) >= 3/2

If a ,b ,c are in G.P. then prove that (a^2+a b+b^2)/(b c+c a+a b)=(b+a)/(c+b)

Prove that: (a+b)^3=a^3+b^3+3a b(a+b)

For a , b in R prove that (|a+b|)/(1+|a+b|) le (|a|)/(1+|a|)+(|b|)/(1+|b|)

If A=2B , then prove that either c=b or a^2 = b(c+b)

Given are positive rational numbers a ,b , c such that a+b+c=1 , then prove that a^a b^bc^c+a^b b^c c^a+ a^c b^a c^b<1.

If a+b=1,a >0, prove that (a+1/a)^2+(b+1/b)^2geq(25)/2dot

If a + b =1, a gt 0,b gt 0, prove that (a + (1)/(a))^(2) + (b + (1)/(b))^(2) ge (25)/(2)