inequality, JMO

How many positive integer values can x take that satisfy the inequality 
(x - 8) (x - 10) (x - 12).......(x - 50) < 0

inequality

Solve the inequality:

(|x|-2)² >= x+4

inequality

Solve the inequality: x²(3 – x)(x – 2)²(x + 4)³ > 0

inequality

Show that: a² + b² + c² ≥ bc + ca + ab

Show that:  a³ + b³ ≥ a²b + ab²

inequality

Show that a⁴ + b⁴ > a³b + ab³ for a ≠b

inequality,

For a given sum of two numbers, when will the product be maximum?

inequality

Show that the sum of a real positive quantity and reciprocal is never less than 2.

inequality

Prove that:

m² + 1/m² > m + 1/m

inequality,

Prove that:

a/b² + b/a² >  1/a + 1/b

Inequality

Prove that:

(a+ b). (b + c). (c + a) ≥ 8abc

Prove that:

(ab + cd).(ac + cd) ≥ 4abcd