MHB What is the expected time difference for two posts in the same minute?

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The discussion centers on finding the sum of all possible values of the expression $\lfloor x\rfloor+\lfloor y\rfloor+\lfloor z\rfloor$ given the conditions $x+y+z=6$ and $xy+yz+xz=9$. The polynomial formed from the roots $x, y, z$ indicates that the only valid values for the expression are 4, 5, and 6, leading to a total sum of 15. Additionally, there is a light-hearted exchange about the timing of posts, suggesting that the expected time difference between two posts in the same minute is less than 60 seconds. The conversation reflects both mathematical inquiry and camaraderie among participants.
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If $x, y, z$ are real numbers such that $x+y+z=6$, $xy+yz+xz=9$, find the sum of all possible values of the expression $\lfloor x\rfloor+\lfloor y\rfloor+\lfloor z\rfloor$.
 
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Re: Evaluate ⌊x⌋+⌊y⌋+⌊z⌋

anemone said:
If $x, y, z$ are real numbers such that $x+y+z=6$, $xy+yz+xz=9$, find the sum of all possible values of the expression $\lfloor x\rfloor+\lfloor y\rfloor+\lfloor z\rfloor$.

Since the floor of a real number is at most, but not quite, 1 point lower than the original number, it follows that:
$$3 < ⌊x⌋+⌊y⌋+⌊z⌋ \le 6$$
$$4 \le ⌊x⌋+⌊y⌋+⌊z⌋ \le 6$$

Working out the equations for instance for x=0, x=ε, and x=1-ε (where ε > 0 is an arbitrary small number), shows that the numbers 4, 5, and 6 are all possible.
Therefore the sum of all possible values of ⌊x⌋+⌊y⌋+⌊z⌋ is 4+5+6=15.
 
Re: Evaluate ⌊x⌋+⌊y⌋+⌊z⌋

[sp]Let $k=xyz$. The polynomial with roots $x,y,z$ is then $\lambda^3 - 6\lambda^2 + 9\lambda - k.$


You can see from the graph that the only values of $k$ for which the equation $k = \lambda^3 - 6\lambda^2 + 9\lambda$ has three real roots are $0\leqslant k\leqslant4.$ As $k$ increases from $0$ to $4$, we can tabulate the values of the roots as follows, where the $+$ and $-$ subscripts mean addtition or subtraction of a small amount (less than $1/2$). $$\begin{array}{c|c|c|c}k&x,y,z & \lfloor x\rfloor,\, \lfloor y\rfloor,\, \lfloor z\rfloor & \lfloor x\rfloor+\lfloor y\rfloor+\lfloor z\rfloor \\ \hline 0& 0,\,3,\,3 &0,\,3,\,3 & 6 \\ 1 & 0_+,\,3_-,\,3_+ & 0,\,2,\,3 & 5 \\ 2 & 0_+,\,2,\,4_- & 0,\,2,\,3 & 5 \\ 3 & 0_+,\,2_-,\,4_- & 0,\,1,\,3 & 4 \\ 4& 1,\,1,\,4 & 1,\,1,\,4 & 6 \end{array}$$ The only possible values for $\lfloor x\rfloor+\lfloor y\rfloor+\lfloor z\rfloor$ are $4$, $5$ and $6$. If I read the question correctly, it asks for the sum of those values, which is $15.$[/sp]

Edit. I like Serena beat me by just seconds!
 

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Re: Evaluate ⌊x⌋+⌊y⌋+⌊z⌋

Opalg said:
Edit. I like Serena beat me by just seconds!

We can only see that we posted in the same minute.

Let $x$ be the time I like Serena posted in minutes, and let $y$ be the time Opalg posted.
Then we know that $⌊x⌋=⌊y⌋$ and also that $x<y$.
Therefore $0 < y-x < 60 \text{ s}$.
Note that a smaller amount is more likely, since with higher amounts the probability increases that we'd have posted in different minutes.
The leaves the question what the expected time difference is. ;)
 
Re: Evaluate ⌊x⌋+⌊y⌋+⌊z⌋

Hey I like Serena and Opalg,

Thank you so so much for participating! At first I thought that folks are jaded with me already...:o getting bored because I posted almost a challenge a day here without fail. To be completely candid, sometimes, I even ask Mark if it's appropriate for me to keep posting!

Solution which I found along with the problem:

$6=x+y+z$

$3=(x-1)+(y-1)+(z-1)<\lfloor x \rfloor+\lfloor y \rfloor+\lfloor z \rfloor \le \lfloor x+y+z \rfloor=6$

$\therefore \lfloor x \rfloor+\lfloor y \rfloor+\lfloor z \rfloor=4, 5, 6$

and hence $\lfloor x \rfloor+\lfloor y \rfloor+\lfloor z \rfloor=4+5+6=15$.

I like Serena said:
We can only see that we posted in the same minute.

Let $x$ be the time I like Serena posted in minutes, and let $y$ be the time Opalg posted.
Then we know that $⌊x⌋=⌊y⌋$ and also that $x<y$.
Therefore $0 < y-x < 60 \text{ s}$.
Note that a smaller amount is more likely, since with higher amount the probability increases that we'd have posted in different minutes.
The leaves the question what the expected time difference is. ;)

I laughed out loud (more than once) when I read this, I like Serena, you have a wonderful personality and a sense of humor!
 
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