Number theory yes no it is Geometry

Geometry Level pending

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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