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1) If $$z = \frac{ \left\{ \sqrt{3} \right\}^2 - 2 \left\{ \sqrt{2} \right\}^2 }{ \left\{ \sqrt{3} \right\} - 2 \left\{ \sqrt{2} \right\} }$$ find $\lfloor z \rfloor$.

2) Find all $x$ for which $$\left| x - \left| x-1 \right| \right| = \lfloor x \rfloor.$$ Express your answer in interval notation.

3) Which positive real number $x$ has the property that $x$, $\lfloor x \rfloor$, and $x - \lfloor x\rfloor$ form a geometric progression (in that order)? (Recall that $\lfloor x\rfloor$ means the greatest integer less than or equal to $x$.)

4) Let $$N = \sum_{k = 1}^{1000}k(\lceil \log_{\sqrt {2}}k\rceil - \lfloor \log_{\sqrt {2}}k \rfloor). $$ Find $N$.

5) Let $f(x)$ be the function whose domain is all positive real numbers defined by the formula $$f(x) = \begin{cases} \dfrac{\sqrt{2x+5}-\sqrt{x+7}}{x-2} & x \neq 2\\ k & x = 2 \end{cases}$$If $f(x)$ is continuous, what is the value of $k$?

6) Prove that $\lfloor 2x \rfloor + \lfloor 2y \rfloor \geq \lfloor x \rfloor + \lfloor y \rfloor + \lfloor x+y \rfloor$ for all real $x$ and $y$.


Thanks for help in advance!

 Jan 20, 2019

Hmm. If I use LaTeX here in the answer some of it displays properly in the question.  If I delete the LaTeX here the question looks a mess again!!


So:  Use the LaTeX button and copy and paste your questions into the resulting box, having removed the $ signs. e.g.


 If \(z = \frac{ \left\{ \sqrt{3} \right\}^2 - 2 \left\{ \sqrt{2} \right\}^2 }{ \left\{ \sqrt{3} \right\} - 2 \left\{ \sqrt{2} \right\} }\) find \(\lfloor z \rfloor\)

 Jan 21, 2019
edited by Alan  Jan 21, 2019
edited by Alan  Jan 21, 2019
edited by Alan  Jan 21, 2019

1.) If, \(z = \frac{ \left\{ \sqrt{3} \right\}^2 - 2 \left\{ \sqrt{2} \right\}^2 }{ \left\{ \sqrt{3} \right\} - 2 \left\{ \sqrt{2} \right\} } \), find \(\lfloor z \rfloor\).


The fact that \(\lfloor \sqrt{2} \rfloor = \lfloor \sqrt{3} \rfloor = 1\), should help. Rewrite the roots in brackets with \(\{\sqrt{2}\} = \sqrt{2} - 1\) and \(\{\sqrt{3}\} = \sqrt{3} - 1\)

 Jan 26, 2019

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