Exercises:
- Show that
is symmetric with respect to a, b, c, d.
- Show that the following expressions are cyclic with respect to a, b, c, d, taken in this order:
and
- Expand the expression using
notation:
- Expand the expression using
notation:
- Prove that
- Prove that
- Prove that
- Prove that :
- Prove that:
- Prove that :
.
- Prove that:
- Prove that:
- Prove that:
- Prove that:
- If one of the numbers a, b, and c is the geometric mean of the other two, use the previous problem to prove the following:
- If the numbers x, y, z taken in some order or other form an AP, use problem 3 to prove that
- Express
as the sum of three squares. Hence, show that
is negative for all real values of a, b, c except when
. Hint: Put
,
,
, and notice that
.
- If
, show that (i)
; (ii)
(iii)
can be expressed in the form
; and, find p, q, r in terms of a, b, c, f, g, h.
Cheers,
Nalin Pithwa.