# Easier Version of an Old HMMT Problem

Algebra Level 5

Let $$a\geq b\geq c$$ be real numbers with $$a+b+c>0$$ such that \begin{align*}a^2bc+ab^2c+abc^2+21&=a+b+c,\\a^2b+a^2c+b^2c+b^2a+c^2a+c^2b+3abc&=-3,\\a^2b^2c+ab^2c^2+a^2bc^2&=7+ab+bc+ca.\end{align*} If $$a^2$$ can be written as $$\dfrac{m+\sqrt n}p$$ for positive integers $$m,n,$$ and $$p$$, find $$m+n+p$$.

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