by 
different methods. The same inequality can be used twice but in different ways, dm me if you have any question about the rules.
be the orthocenter of acute triangle 
,
be the foot of the altitude from 
to 
,
be the reflection of 
.
intersects line 
and 
.
.
for which the sum ![\[U=\sum_{n=1}^{2023}\left\lfloor\dfrac{n^{2}-na}{5}\right\rfloor\]](http://latex.artofproblemsolving.com/6/b/0/6b010978a55642b7f7aabb2df2f7cf45c9db937a.png)
is an integer strictly between 
and 
.
.
denotes the greatest integer that is less than or equal to 
.
weights, each weighing 
and another 
weights with 
each. Write a algorithm that determines the ways in which a scale can be balanced with a weight 
.
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Dear AMC Students:
There seems to be some confusion about the meaning of one of our notes to contest managers, about changes of venue from AMC10/12 to AIME. Please know that we are committed to making it possible for all qualified AIME candidates to take the contest.
It has always been our policy that students should take the two contests in the same place, that this place should be a school or other public facility, and that the contest manager (proctor) not be a family member of any candidate.
These rules have not changed, and we ask you to honor them with rare exception. If your original contest site is unavailable to you (weather, vacation, local holiday), or if you are travelling during the AIME administration day, you must find an alternative venue which fits our requirements and which is willing to administer to you the AIME contest. Then you should have the contest manager (proctor) at the alternative venue write us an email at amcinfo@maa.org . Ours is a school-based program. We will arrange for contest administration with the personnel at the alternative venue.
We ask that you refrain from changing venue simply because of convenience. It takes hours, without exaggeration, to effect even a few changes of venue. We offer the AIME at no cost to the field, and the effort made by the mathematical community to sustain this effort is considerable. We ask you, as future members of this community, to take seriously this effort, by thinking two or three times before initiating a change of venue request.
While I have your ear, you can help in another way: you can remember the name you used in writing the AMC10/12 contest, and use exactly the same name for the AIME contest. Following this rule, and also the rules for change of venue, makes it easier for us to offer you the USA(J)MO, should you qualify.
We thank you in advance for your cooperation in this situation.
There seems to be some confusion about the meaning of one of our notes to contest managers, about changes of venue from AMC10/12 to AIME. Please know that we are committed to making it possible for all qualified AIME candidates to take the contest.
It has always been our policy that students should take the two contests in the same place, that this place should be a school or other public facility, and that the contest manager (proctor) not be a family member of any candidate.
These rules have not changed, and we ask you to honor them with rare exception. If your original contest site is unavailable to you (weather, vacation, local holiday), or if you are travelling during the AIME administration day, you must find an alternative venue which fits our requirements and which is willing to administer to you the AIME contest. Then you should have the contest manager (proctor) at the alternative venue write us an email at amcinfo@maa.org . Ours is a school-based program. We will arrange for contest administration with the personnel at the alternative venue.
We ask that you refrain from changing venue simply because of convenience. It takes hours, without exaggeration, to effect even a few changes of venue. We offer the AIME at no cost to the field, and the effort made by the mathematical community to sustain this effort is considerable. We ask you, as future members of this community, to take seriously this effort, by thinking two or three times before initiating a change of venue request.
While I have your ear, you can help in another way: you can remember the name you used in writing the AMC10/12 contest, and use exactly the same name for the AIME contest. Following this rule, and also the rules for change of venue, makes it easier for us to offer you the USA(J)MO, should you qualify.
We thank you in advance for your cooperation in this situation.
