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Let \( r_1, r_2, r_3\) be the roots of polynomial \( P(x) = x^3 + 3x + 1\). Evaluate the product \[ \displaystyle \prod_{k=1}^3 (r_k^2 + r_k + 1).\]

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Let \(N(k)\) be equal to the \(21^{\text{st}}\) derivative of \(k^3 \sin k^2\). Determine the sum of the digits of \(N(0)\).

If 271 is written as the sum of positive real numbers so as to maximize the product of the summands, how many summands would be there?

As an example: 271 ...

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