A Not-So-Special Cubic Function

Algebra Level 5

Let \(f(x)\) be a monic cubic function with the following properties:

  • For any line \(L(x)\) such that \(L(2)=2\) that intersects \(f(x)\) at three distinct points \((x_{1},y_{1}),(x_{2},y_{2}),(x_{3},y_{3}),\) the sum \(x_{1}+x_{2}+x_{3}=2\).

  • For any quadratic function \(P(x)\) such that \(P(0)=0\) that intersects \(f(x)\) at three distinct points \((x_{4},y_{4}),(x_{5},y_{5}),(x_{6},y_{6}),\) the product \(x_{4}x_{5}x_{6}=-3\).

  • \(f(2)=17\)

Find the value of \(f(5)\).

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