,
and 
meet at a point 
.
and the points 
lie on the same straight line.
be real numbers such that 
Prove that
.
are there such that 
and 
is an integer.
be reals such that 
. Prove that
Let 
. Prove that
and 
for each positive integer 
,
?
and ![$e\in [-2,2]$](http://latex.artofproblemsolving.com/b/b/7/bb78c9dedb18af5e6a904f5492917921e41a7ebc.png)
such that 
Find the value of 
be a positive integer such that 
,![\[
A = \{a_1, a_2, a_3, \ldots, a_m\}
\]](http://latex.artofproblemsolving.com/1/7/9/179687a134d5495ffff0b44fe763a2bb8b3ddd9c.png)
consist of distinct positive integers not exceeding 2025. Suppose that for every 
,
,
,
as well. Prove that![\[
\frac{a_1 + a_2 + a_3 + \cdots + a_m}{m} \geq 1013.
\]](http://latex.artofproblemsolving.com/c/0/d/c0d190f78d94deac1b947ba57a2de51d0b06f0bd.png)
for which I can find nonnegative integers 
such that for all indices 
,
divides 
?
...")
integers - take the set of all integers with a digit sum of 
in base 
to the list.
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20th ELMO on AoPS: Eyy LMaO
Jun 9th and 16th, 2018
It's me again! Here for the big two-zero!
The ELMO is an olympiad test similar to both the USAMO and IMO in format. It is written, administrated and graded by returning MOPers for those attending MOP for the first time. However, because there are only finitely many people who can attend MOP each year, for many years the competition has also been posted and run on AoPS, similarly to many of the other mocks that run on this site. Here are to the previous AoPS ELMO threads. You can also find the problems and shortlist in the USA contests section of the Contest Collections. The acronym is different every year, and will be chosen during the first week of MOP.
Jun 9th and 16th, 2018
It's me again! Here for the big two-zero!
The ELMO is an olympiad test similar to both the USAMO and IMO in format. It is written, administrated and graded by returning MOPers for those attending MOP for the first time. However, because there are only finitely many people who can attend MOP each year, for many years the competition has also been posted and run on AoPS, similarly to many of the other mocks that run on this site. Here are to the previous AoPS ELMO threads. You can also find the problems and shortlist in the USA contests section of the Contest Collections. The acronym is different every year, and will be chosen during the first week of MOP.
- You should sign up in this thread if you intend to do the contest. Signups are neither mandatory nor binding, but we want to have an estimate of how many people take the test.
- You may participate in either or both of the two days. Each day will consist of three problems, of a similar difficulty as the USAMO. The top scores from each day and overall will be recognized, and all scores will be posted on the ELMO website, where you can also find past results.
- Solutions will be due on Wednesday, June 27th at 11:59 PM PDT.
- Each day should be taken in a contiguous 4.5-hour period, similarly to how you would take the USAMO, although you do not need a proctor and you may take the test anytime before the submission deadline.
- Submit your solutions by PM to whatshisbucket and suli on AoPS or to elmograding2018@gmail.com. You may either scan written solutions or write your solutions in LaTeX. You may write your solutions on paper and then transcribe them to LaTeX immediately afterwards, provided that you do so entirely verbatim. Please only do so if you have hideous handwriting like mine that only you can decipher or do not have anything resembling a scanner available.
- Please do not discuss the problems with anyone else before you take the test. After you have taken the test, please do not discuss the problems with anyone who has not yet taken the test. After the tests are over, the problems will be posted in the High School Olympiads forum.
We look forward to your participation!
This post has been edited 11 times. Last edited by whatshisbucket, May 10, 2019, 12:53 AM
got 1 point, proof of 
,
equal gets 6 points. Do not attempt to convert your 6 to a 7 because you overlooked the obvious.
