Means, Monics and More

Level pending

The arithmetic mean, the geometric mean and the harmonic mean of the roots of the polynomial \(\begin{align} P(x) = x^{100} - 2 x^{99} + 3 x^{98} - 4 x^{97} + (\cdots) - 100x + 101 \end{align}\) are the only roots of the monic polynomial \(Q(x)\).

Evaluate the sum of all true statements.

(0) = There are no true statements.

(1) = \(AM > GM > HM\).

(2) = \(P(x) > Q(x), \: x \in \mathbb{R}\).

(4) = \(Q(0) > 2\).

(8) = \(P(x)\) has no real roots.

(16) = \(Q(Q(x))\) has 6 real roots.

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