# Number theory yes no it is Geometry

Geometry Level 5

Let $$m$$ and $$n$$ be positive integers satisfying $$n^5 + 3n + 4 = 2^m$$. Let the sum of all possible values of $$n$$ be $$A$$.

Let the sum of all positive integers $$x$$ for which $$\lfloor A/4 \rfloor ( \lfloor A/2 \rfloor n + A - 1) (An + 2A - 1) + 9$$ is a perfect cube of a positive integer is $$B$$.

$$Then\quad are\quad given\quad three\quad \quad lines\quad a,b\quad and\quad c\quad and\quad no\quad three\quad of\\ them\quad intersect\quad in\quad one\quad point,\quad and\quad no\quad two\quad of\quad them\\ \quad are\quad parallel,and\quad a\cap b=A,b\cap c=B,c\cap a=C.\\ The\quad tree\quad circles\quad that\quad are\quad tangent\quad to\quad \quad a,b\quad and\\ \quad c\quad and\quad are\quad not\quad in\quad the\quad interior\quad of\quad ABC\quad have\quad radii\quad \\ A+2B,|A-2B|,2A\times B.If\quad the\quad perimeter\quad of\quad ABC\quad is\quad \quad k\sqrt { l } ,\\ where\quad k\quad and\quad l\quad are\quad positive\quad integers\quad and\quad l\quad is\quad square\quad free.\\ Then\quad find\quad k+l!$$

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