[/quote]
,
reals, and 
.
?
be positive real numbers such that 
.
knowing that the denominators are positive real numbers.
such that: Among every 
such that a positive integer power of 
equals to a positive integer power of 
.
be an acute-angled triangle with 
A circle passes through 
and 
and crosses the segments 
and 
again at 
and 
respectively. The circumcircles of 
and 
and 
The segments 
and 
intersect at 
Let 
and 
be the reflections of 
in the lines 
and 
respectively. Prove that the points 
and 
are concyclic.
be positive real numbers such that 
.
When is the equality valid?
be a center of circle which passes through vertices of quadrilateral 
,
of positive integers such that ![\[ k!=(2^n-1)(2^n-2)(2^n-4)\cdots(2^n-2^{n-1}). \]](http://latex.artofproblemsolving.com/3/5/3/353bbb15bac205053c26208af8cac7c32a296f55.png)
Proposed by Gabriel Chicas Reyes, El Salvador
cards on the table, and each card has a positive integer written on it. An odd number is written on exactly 
cards. Every minute, the following operation is performed: for all possible sets of 
cards on the table, the product of the numbers on these three cards is calculated, all the obtained results are summed, and this sum is written on a new card and placed on the table. A day later, it turns out that there is a card on the table, the number written on this card is divisible by 
Prove that one hour after the start of the process, there was a card on the table that the number written on that card is divisible by 
on the board. Charlotte has two operations available: the GCD operation and the LCM operation.
and 
written on the board, erasing them, and writing the integer 
.
.
is called a winning number if there exists a sequence of operations such that, at the end, the only integer left on the board is 
and, for each of them, determine the minimum number of GCD operations Charlotte must use.
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Greetings, AoPS users!
If you're reading this post, that means you've come across some kind of bug, error, or misbehavior, which nobody likes! To help us developers solve the problem as quickly as possible, we need enough information to understand what happened. Following these guidelines will help us squash those bugs more effectively.
Before submitting a bug report, please confirm the issue exists in other browsers or other computers if you have access to them.
Please do not report the following known issues unless you are able to tell us in detail how to recreate the issue:
For a list of many common questions and issues, please see our user created FAQ, Community FAQ, or For the Win! FAQ.
What is a bug?
A bug is a misbehavior that is reproducible. If a refresh makes it go away 100% of the time, then it isn't a bug, but rather a glitch. That's when your browser has some strange file cached, or for some reason doesn't render the page like it should. Please don't report glitches, since we generally cannot fix them. A glitch that happens more than a few times, though, could be an intermittent bug.
If something is wrong in the wiki, you can change it! The AoPS Wiki is user-editable, and it may be defaced from time to time. You can revert these changes yourself, but if you notice a particular user defacing the wiki, please let an admin know.
The subject
The subject line should explain as clearly as possible what went wrong.
Bad: Forum doesn't work
Bad: Question
Good: Switching between threads quickly shows blank page.
We may remove posts for which the topic does not easily identify the issue. Topics named such as "Question", "Weird issue," "Why?" and the like will likely be removed.
The report
Use this format to report bugs. Be as specific as possible. If you don't know the answer exactly, give us as much information as you know. Attaching a screenshot is helpful if you can take one.
Summary of the problem:
Page URL:
Steps to reproduce:
1.
2.
3.
...
Expected behavior:
Frequency:
Operating system(s):
Browser(s), including version:
Additional information:
If your computer or tablet is school issued, please indicate this under Additional information.
Example
Summary of the problem: When I click back and forth between two threads in the site support section, the content of the threads no longer show up. (See attached screenshot.)
Page URL: http://artofproblemsolving.com/community/c10_site_support
Steps to reproduce:
1. Go to the Site Support forum.
2. Click on any thread.
3. Click quickly on a different thread.
Expected behavior: To see the second thread.
Frequency: Every time
Operating system: Mac OS X
Browser: Chrome and Firefox
Additional information: Only happens in the Site Support forum. My tablet is school issued, but I have the problem at both school and home.
How to take a screenshot
Mac OS X: If you type ⌘+Shift+4, you'll get a "crosshairs" that lets you take a custom screenshot size. Just click and drag to select the area you want to take a picture of. If you type ⌘+Shift+4+space, you can take a screenshot of a specific window. All screenshots will show up on your desktop.
Windows: Hit the Windows logo key+PrtScn, and a screenshot of your entire screen. Alternatively, you can hit Alt+PrtScn to take a screenshot of the currently selected window. All screenshots are saved to the Pictures → Screenshots folder.
Advanced
If you're a bit more comfortable with how browsers work, you can also show us what happens in the JavaScript console.
In Chrome, type CTRL+Shift+J (Windows, Linux) or ⌘+Option+J (Mac).
In Firefox, type CTRL+Shift+K (Windows, Linux) or ⌘+Option+K (Mac).
In Internet Explorer, it's the F12 key.
In Safari, first enable the Develop menu: Preferences → Advanced, click "Show Develop menu in menu bar." Then either go to Develop → Show Error console or type Option+⌘+C.
It'll look something like this:
If you're reading this post, that means you've come across some kind of bug, error, or misbehavior, which nobody likes! To help us developers solve the problem as quickly as possible, we need enough information to understand what happened. Following these guidelines will help us squash those bugs more effectively.
Before submitting a bug report, please confirm the issue exists in other browsers or other computers if you have access to them.
Please do not report the following known issues unless you are able to tell us in detail how to recreate the issue:
- Avatar or user appearing twice in topic or private message. Please just refresh.
- Friends showing twice. Please refresh.
- Not being able to search a forum after going back a page.
- The edit icon showing for a message that is not yours.
Please also do not post issues regarding problems in your homework, Alcumus, For the Win!, or MATHCOUNTS Trainer. Please use the corresponding report problem features of the applications. Site Support staff are unable to help with these issues.
For a list of many common questions and issues, please see our user created FAQ, Community FAQ, or For the Win! FAQ.
What is a bug?
A bug is a misbehavior that is reproducible. If a refresh makes it go away 100% of the time, then it isn't a bug, but rather a glitch. That's when your browser has some strange file cached, or for some reason doesn't render the page like it should. Please don't report glitches, since we generally cannot fix them. A glitch that happens more than a few times, though, could be an intermittent bug.
If something is wrong in the wiki, you can change it! The AoPS Wiki is user-editable, and it may be defaced from time to time. You can revert these changes yourself, but if you notice a particular user defacing the wiki, please let an admin know.
The subject
The subject line should explain as clearly as possible what went wrong.
Bad: Forum doesn't work
Bad: Question
Good: Switching between threads quickly shows blank page.
We may remove posts for which the topic does not easily identify the issue. Topics named such as "Question", "Weird issue," "Why?" and the like will likely be removed.
The report
Use this format to report bugs. Be as specific as possible. If you don't know the answer exactly, give us as much information as you know. Attaching a screenshot is helpful if you can take one.
Summary of the problem:
Page URL:
Steps to reproduce:
1.
2.
3.
...
Expected behavior:
Frequency:
Operating system(s):
Browser(s), including version:
Additional information:
If your computer or tablet is school issued, please indicate this under Additional information.
Example
Summary of the problem: When I click back and forth between two threads in the site support section, the content of the threads no longer show up. (See attached screenshot.)
Page URL: http://artofproblemsolving.com/community/c10_site_support
Steps to reproduce:
1. Go to the Site Support forum.
2. Click on any thread.
3. Click quickly on a different thread.
Expected behavior: To see the second thread.
Frequency: Every time
Operating system: Mac OS X
Browser: Chrome and Firefox
Additional information: Only happens in the Site Support forum. My tablet is school issued, but I have the problem at both school and home.
How to take a screenshot
Mac OS X: If you type ⌘+Shift+4, you'll get a "crosshairs" that lets you take a custom screenshot size. Just click and drag to select the area you want to take a picture of. If you type ⌘+Shift+4+space, you can take a screenshot of a specific window. All screenshots will show up on your desktop.
Windows: Hit the Windows logo key+PrtScn, and a screenshot of your entire screen. Alternatively, you can hit Alt+PrtScn to take a screenshot of the currently selected window. All screenshots are saved to the Pictures → Screenshots folder.
Advanced
If you're a bit more comfortable with how browsers work, you can also show us what happens in the JavaScript console.
In Chrome, type CTRL+Shift+J (Windows, Linux) or ⌘+Option+J (Mac).
In Firefox, type CTRL+Shift+K (Windows, Linux) or ⌘+Option+K (Mac).
In Internet Explorer, it's the F12 key.
In Safari, first enable the Develop menu: Preferences → Advanced, click "Show Develop menu in menu bar." Then either go to Develop → Show Error console or type Option+⌘+C.
It'll look something like this:
This post has been edited 14 times. Last edited by levans, Apr 16, 2017, 6:00 AM
