Algebra

[sparky69er]

Let a, b, c be positive real unequal numbers.

Using the results 1) a^2 + b^2 > 2ab
2) b^2 + c^2 > 2bc
3) c^2 + a^2 > 2ac

Deduce that a^2 - ab + b^2 > ab

Deduce that a^2 + b^2 + c^2 > bc + ac + ab

Show that a^3 + b^3 > ab(a + b)

[Abbas Haider]

I will help you with yur first prob whenever I will figure out the second and third one I will post it to u

Problem 1:
Since
a^2+b^2>2ab
a^2+b^2>ab+ab
a^2+b^2-ab>ab

[Logophobic]

2) Add all three inequalities:
(a² + b²) + (b² + c²) + (c² + a²) > 2ab + 2bc + 2ac
2(a² + b² + c²) > 2(ab + bc + ac)
a² + b² + c² > ab + bc + ac


3)Find inequalities for a³ and b³ separately, then add them together:
a² + b² > 2ab
a² > 2ab - b²
a³ > 2a²b - ab²
Similarly:
b³ > 2ab² - a²b
Thus:
a³ + b³ > a²b + ab²
a³ + b³ > ab(a + b)

[Abbas Haider]

Nice going logophobic