[/quote]
,
(i dont remember exactly if it was 8032038 or some other number, so correct me on this)
(i posted this at 12:29) lol
?
be a function satisfying![\[
f(f(x)) = x^2 - 6x + 6
\quad \text{for all} \quad x \in \mathbb{Z}.
\]](http://latex.artofproblemsolving.com/8/a/e/8aebed1e260316d675ca7d104d83fabcd8b5717e.png)
Given that![\[
f(i) < f(i+1) \quad \text{for} \quad i = 0, 1, 2, 3, 4, 5,
\]](http://latex.artofproblemsolving.com/9/f/9/9f92574f9fe14e2e30c3e51e194c661fbd4ece07.png)
find the value of![\[
f(0) + f(1) + f(2) + \cdots + f(6).
\]](http://latex.artofproblemsolving.com/4/0/7/4075d7efa78dda4c46a5cf764fb8852e8c0fb09b.png)
such that for all 
the followings hold:
be a triangle with circumcentre 
.
lie in the interiors of the sides 
respectively, such that 
is perpendicular to 
and 
is perpendicular to 
.
lies on the line 
and 
and 
on that line.)
be the circumcentre of triangle 
.
and 
?
of positive integers, such that every sum
(with 
)
for all 
;
for all 
?
denotes the greatest common divisor of 
,
.
such that ![\[ f(x^2 + yf(z)) = xf(x) + zf(y) , \]](http://latex.artofproblemsolving.com/2/b/a/2baee1d8464992fc38a16561b48b18f4b35807bd.png)
for all 
.
,
,
be an infinite sequence of positive integers. Suppose that there is an integer 
such that, for each 
,
is an integer. Prove that there is a positive integer 
such that 
for all 
.
,
,
be positive real numbers such that 
.![$$\sqrt[3]{(a^2+1)(b^2+1)(c^2+1)} \le \left(\frac{a+b+c}{3}\right)^2 + 1.$$](http://latex.artofproblemsolving.com/2/0/7/207934bca53047858a3820f37705171ff0abfd45.png)
and 
and 
Prove that![\[\frac{a}{(b+c-a)^2} + \frac{b}{(c+a-b)^2} + \frac{c}{(a+b-c)^2} \geq \frac{3}{(abc)^2}.\]](http://latex.artofproblemsolving.com/0/a/b/0ab4b4e6c28a1ae67788525642661dbcd04d0bd8.png)
,
.
be the midpoints of 
and 
,
is acute, and let 
be its orthocentre. Points 
and 
lie on lines 
and 
,
and 
and 
and 
Y by kk108, Mudkipswims42, atmchallenge, MSTang, hwl0304, trumpeter, psa, rkm0959, 62861, anantmudgal09, Ankoganit, droid347, wu2481632, champion999, TheOneYouWant, talkon, laegolas, eisirrational, something_boring, checkmatetang, ThisIsASentence, iks92, itised, zxcv1337, soojoong, v_Enhance, AOPS12142015, Mathisfun04, Plasma_Vortex, artsolver, JasperL, huricane, pi_Plus_45x23, Problem_Penetrator, OrionNebula, GeneralCobra19, bogstop320, mathwizard888, DaniyalQazi2, Pure_IQ, blep, umaru, vjdjmathaddict, shinichiman, RYang, strategos21, kingofgeedorah, joey8189681, Marzi, NOLF, tapir1729, e_plus_pi, mathleticguyyy, amar_04, OlympusHero, Adventure10, Mango247
19th ELMO on AoPS: The vEry badLy naMed cOntest
Jun 10th and 17th, 2017
Results!
Day 2 has been released!
Both days are attached at the bottom of this post.
Hi! Since v_Enhance and ABCDE are both busy being at MOP this year, I will being running this year's ELMO on AoPS.
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 10th and 17th, 2017
Results!
Day 2 has been released!
Both days are attached at the bottom of this post.
Hi! Since v_Enhance and ABCDE are both busy being at MOP this year, I will being running this year's ELMO on AoPS.
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. You are not required to sign up in order to take it, nor will you be required to take it if you sign up, but we want to have an estimate of how many people take the test and engage in some of the same tomfoolery as last year.
- 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 June 25th.
- 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.
- Submit your solutions to whatshisbucket, trumpeter, and suli on AoPS. 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 10 times. Last edited by whatshisbucket, Jun 28, 2017, 3:49 AM
for 
and 
