Chứng minh

To prove the inequality, we can start by expanding both sides:

m + n + p^2 >= 3mn + 3np + 3pm

Rearranging the terms, we have:

m + n + p^2 - 3mn - 3np - 3pm >= 0

Now, let's factor out common terms:

m(1 - 3n - 3p) + n(1 - 3m - 3p) + p^2 - 3np >= 0

Next, we can simplify the expressions inside the parentheses:

1 - 3n - 3p = -(3n + 3p - 1)
1 - 3m - 3p = -(3m + 3p - 1)

Substituting these back into the inequality, we have:

m(-(3n + 3p - 1)) + n(-(3m + 3p - 1)) + p^2 - 3np >= 0

Expanding further:

-3mn - 3mp + m + 3mn - 3np + n + p^2 - 3np >= 0

Simplifying:

m + n + p^2 - 3mp - 3np >= 0

Now, let's rearrange the terms:

m - 3mp + n - 3np + p^2 >= 0

Factoring out common terms:

m(1 - 3p) + n(1 - 3p) + p^2 >= 0

Simplifying further:

(1 - 3p)(m + n) + p^2 >= 0

Since (m + n) and p^2 are both non-negative, the inequality holds true for all real values of m, n, and p.