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Given that $\{x\}$, $\lfloor x \rfloor$, and $x$ form a geometric progression, find $x$.

Notations:

Suppose the set $S=\{ cat,dog,potato\}$, when equipped with a binary operation $\cdot$, forms a group.

You know that $dog \cdot dog = potato$.

What is $cat \cdot dog$ ?

For each positive integer $n$, $S_n$ is the sum of the first $n$ consecutive prime numbers.

For example:

...

Given a positive infinite sequence $\{a_n\}$, $S_n = \displaystyle \sum_{k=1}^{n} a_k$.

If $\forall n \in \mathbb N^+$, the arithmetic mean of $a_n$ and $2$ is equal to the geometric mean …

$x \left(8 \sqrt{1- x}+ \sqrt{1 + x}\right) ≤ 11 \sqrt{1 + x} - 16 \sqrt{1 - x}$

$$

Solve the inequality above for positive $x$.

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