practice problems pending

 Q1.  Let (x, y, z) be positive real numbers such that xy + yz + zx = 3. Prove that:
(x^2 + y^2)/z + (y^2 + z^2)/x + (z^2 + x^2)/y >= 6.
S1.

AM GM on x^2/z and z => x^2/z + z >= 2x
=> LHS + 2(x+y+z) >= 4(x+y+z)
=> LHS  >= 2(x+y+z)

Using xy + yz + zx = 3 and Rearrangement inequality:
(x+y+z)^2 >= 3(xy + yz + zx) = 9
=> x + y + z >= 3
H.P.

Q2. Let (a, b, c) be positive real numbers such that abc = 1. Prove that
a/b + b/c + c/a >= a + b + c.
S2.

Using abc =1given LHS = a^2.c + b^2.a + c^2.b

AM GM on  a^2.c and 1/c => a^2.c + 1/c >= 2a

Similarly do others and add all to get:

LHS + 1/a + 1/b + 1/c >= 2(a+b+c)_______[1]

Now do AM GM on
a/b, a/b, c/a
=> 2a/b + c/a >= 3(ac/b^2)^1/3
Using abc = 1
2a/b + c/a >= 3/b

Similarly do others and get:
2b/c + a/b >= 3/c

and

2c/a + b/c >= 3/a

Add all => 3.LHS >= 3(1/a + 1/b + 1/c) => LHS >= sigma(1/a) => LHS + LHS >= LHS + sigma(1/a)
=> 2LHS >= LHS + sigma(1/a)
Using [1] => 2LHS >= 2(a+b+c) H.P.