[/quote]
,
be a cyclic quadrilateral, and let 
,
,
,
be the midpoints of 
,
,
,
respectively. Let 
,
,
and 
be the orthocenters of triangles 
,
,
and 
,
have the same area.
with real coefficients is the average of two monic polynomials of degree 
of positive integers, for which![$$(a, b) + 3[a, b] = a^3 - b^3$$](http://latex.artofproblemsolving.com/b/8/3/b83f52a0152dc39b6d966880a1d6ca824a0f3786.png)
denotes the greatest common divisor of ![$[a, b]$](http://latex.artofproblemsolving.com/d/a/2/da2e551d2ca2155b8d8f4935d2e9757722c9bab6.png)
denotes the least common multiple of 
is an acute angle triangle such that 
and 
.
the center of the circumscribed circle of the triangle and 
intersects 
and 
in point 
.
.
.
be a point on side 
such that 
.
,
satisfying 
.
of positive integers with 
prime and ![\[ a^p=b!+p. \]](http://latex.artofproblemsolving.com/9/f/6/9f674dabcb364872cef4153198e2e362ae40a2fd.png)
rows of seats with 
seats in each row. Eight passengers have boarded the plane and are distributed randomly among the seats. A married couple is next to board. What is the probability there will be 2 adjacent seats in the same row for the couple?
feet high and the other peak is 
feet high. Each peak forms a 
angle, and the straight sides of the mountains form 
with the ground. The artwork has an area of 
square feet. The sides of the mountains meet at an intersection point near the center of the artwork, 
feet above the ground. What is the value of 
satsfying 
and:![\[ f(2n)f(2n+1)=9f(n)^2+3f(n)\]](http://latex.artofproblemsolving.com/b/c/1/bc18d6ad72589f012dcab7836228b3ed7ee061c2.png)
be positive reals such that 
.![\[18\sum_{\text{cyc}}\frac{1}{(3-c)(4-c)}+2(ab+bc+ca)\ge 15. \]](http://latex.artofproblemsolving.com/5/d/5/5d5c47a52d2d6ea7d13ce1c2fa14a08425bbfa5d.png)
Proposed by David Stoner
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 
