Summary: Proof of (a+b+c)/3 is greater than or equal to ((a^2+b^2+c^2)/3)^(1/5) and Schur's inequality

Summary: The given text consists of two parts. The first part contains two examples along with their proofs concerning inequalities. In the first example, it is proved that if 'a_i' are real numbers such that the sum of their squares, the summation of (a_i^2), is less than or equal to 4, then the sum of their cubes, the summation of (a_i^3), will be less than or equal to 8. The proof involves utilizing the AM-GM inequality to demonstrate that the sum of the cubes is bounded by 8. In the second example, an inequality is proved for non-negative numbers 'a', 'b', and 'c': a^3 + b^3 + c^3 - 3abc is greater than or equal to 2((b+c)/2 - a)^3.