and 
Prove that
and 
be positive real numbers with 
.
For which 
such that 
for all 
,
.
and 
of an equilateral triangle 
are taken points 
and 
,
, points 
and 
so that 
.
and 
is equal to 
.
over 
.
![\[\lfloor x \rfloor\]](http://latex.artofproblemsolving.com/d/3/f/d3f8facf62413e1f17c89b9c183a8735c2defe8b.png)
denote the greatest integer less than or equal to x . What is the sum of the squares of the real numbers x for which ![\[ x^2 - 20\lfloor x \rfloor + 19 = 0 \]](http://latex.artofproblemsolving.com/f/a/9/fa96f2d6217a16029e5f03499e0a83deffdfa28a.png)
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(This is an addition to AIME15's post in another forum, and is a more modernized version of rrusczyk's post)
I'm not accusing anybody in particular of any of the following!
Being on Topic
Thread Title
Posting the Problem
What You Tried
Questions
Replies
Basic Etiquette
If you feel like the problem is too easy or hard, please use the Too Easy/Hard buttons (yes, self explanatory). levans spent quite a bit of time developing that, and it would be great if the community adopted it!
Here is the list of your moderators, always ready to help.
If you have any questions, please feel free to private message one of us.
I'm not accusing anybody in particular of any of the following!
Being on Topic
- Be sure to stay on topic in each specific thread; I've personally seen threads deviating greatly off topic. For example, a topic on Simple Arithmetic should not end in a display of Mathematica usage.
- Don't bother other people about straying off topic, the moderators will take note. (Some call this backseat moderation).
Thread Title
- Make your title fairly descriptive and concise. A good example is "MATHCOUNTS Number Theory"; a bad example is "HELP ME!!!!!!!!!!!!!!!!!!!!!!!
- Post the source of the problem.
Posting the Problem
- Use proper spelling and grammar (to the best of your abilities). People who are here to help you out don't want to wade through something like "if 3x/4-x+18=2 what is x".
- Attach a diagram when applicable.
- Search the problem before you post it.
What You Tried
- Post what you attempted, even if it may not have worked. We are not here to give you solutions, we are here to help you learn.
- Be detailed.
Questions
- Ask specific questions; it makes it easier for those helping you.
Replies
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- If you disagree with a reply, cite examples or give some sort of analysis supporting why you disagree.
Basic Etiquette
- Don't make a post to thank somebody who has helped you; even though it is respectful, it clutters up the forum. Use the brand new thank function!
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- Please do not private message moderators with math related questions - those belong on the forums. (Note: I speak only for myself.)
If you feel like the problem is too easy or hard, please use the Too Easy/Hard buttons (yes, self explanatory). levans spent quite a bit of time developing that, and it would be great if the community adopted it!
Here is the list of your moderators, always ready to help.
If you have any questions, please feel free to private message one of us.
