table filled with natural numbers, we say it is a divisor table if:
-
,
-
,
for every 
.
is given. Determine the smallest natural number 
,
be integers so that the integers 
are divisible by 
be a fixed integer. There are 
beads on a circular necklace. You wish to paint the beads using 
consecutive beads every color appears at least once. Find the largest value of 
for which this task is 
possible.
be a triangle 
)
are the altitudes of the triangle. The bisector of 
intersects 
at 
.
be a point such that 
and 
.
passes through the midpoint of 
.
![$f\in C[0,b]$](http://latex.artofproblemsolving.com/8/9/8/898aec0300888e562794b7a551be34f5c2676236.png)
,
and let 
be periodic with period 
.
has a limit as 
and
;
for some real coefficients 
.
for all integers 
such that 
.
for which 
.
where 
,
,
are nonnegative integers.
,
,
and 
for 
.
and give an example where 
.
on 
for which![\[
\lim_{R \to \infty} \int_{\|x\|_2 \leq R} f(x) \, dx = 0
\]](http://latex.artofproblemsolving.com/4/6/9/469f69ee93ae8d1de73c31ceb066253751aa9775.png)
and for which![\[
\lim_{R \to \infty} \int_{\|x\|_\infty \leq R} f(x) \, dx \text{ does not exist.}
\]](http://latex.artofproblemsolving.com/d/f/a/dfadbdd902375b9f67cd2dd0dbb494978bd9e75c.png)
be the set of all 
matrices with at most two entries in each row equal to 
and all other entries equal to 
.
.
denotes the determinant of the matrix 
is differentiable and 
Show that for some 
![$f:[0,1]\to\mathbb R$](http://latex.artofproblemsolving.com/e/a/0/ea02b990406104357c11912a8bf57bea7eeb519d.png)
is differentiable with 
If ![$|f'(x)|\le f(x)\;\forall\;x\in[0,1],$](http://latex.artofproblemsolving.com/b/9/d/b9d6088657cd85b37190b98abac1d94a18bb5916.png)
then show that 
be the unit circle in the complex plane. Let 
be the map given by 
We define 
and 
for 
The smallest positive integer 
is called period of 
Determine the total number of points 
of period 
denote the set of natural numbers, and let 
be nine distinct tuples in 
Show that there are 
distinct elements in the set 
whose product is a perfect cube.
and let 
be positive integers such that 
Prove that 
and determine when equality holds.
be real numbers. Define 
and 
.
and hence find the limit.
,
.
satisfies the quadratic equation 
and deduce that the value of 
.
.
,
is 
.
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TL;DR: MO 3 and new experimental JMO, can take both, sign-up link, problems posted 19 and 20 March, deadline 8 April. Have fun!
What's New
Twelve new problems. Twelve? Yes, in addition to the MO 3, there will also be an experimental JMO. 6+6-12=0, so the MO 3 and the experimental JMO will be disjoint, meaning that you can take both of them! (read: we hope you take both)
About Us
We are a group of Thai students sharing the hobby of proposing olympiad math problems. Most of us had been in the Thailand TST Camp, and several of us are IMO medalists. Some of our members are @ThE-dArK-lOrD and @TacH, who will be co-hosting the MO 3 with me. We have also just launched our website (still kinda work-in-progress): https://www.infinitydots.org.
For the past two years, we have been publishing our problems in InfinityDots MOs—hence the number 3 this year.
MO 1 (2017): main thread, problems
MO 2 (2018): announcement, main thread, problems
Formal Stuff
- Sign-ups are now open in a separate thread (so we don't flood HSO). If you sign up, (i) we’ll notify you when the contest starts, and (ii) it’ll bump that thread up. There’s also the option to privately sign-up—just PM me.
- Both the MO 3 and the experimental JMO will start (i.e. day 1 problems posted) on 19 March at 10am EDT, day 2 problems will be posted 20 March 10am EDT.
- Window for submitting solutions lasts until 8 April 10pm EDT. Scores will be released around a week after that (but definitely before the USA(J)MO) depending on how fast we can grade and how many people submit. Submissions can be done in two ways: either PM me (@talkon) here on AoPS, or send an email to mo3@infinitydots.org.
FAQs
Q: Is the sign-up list for both the MO 3 and the JMO?
A: Yeah, the same sign-up list is used for both the MO 3 and the JMO. Which one (or two) you will take is your choice. You can take only the MO, only the JMO, or both!
Q: Are there any awards?
A: Sadly, no, but we hope you enjoy the experience of solving our problems.
Q: Anything I can do to help you?
A: We are not accepting any testsolvers or graders publicly, but feel free to help us spread the word.
If you have more questions, feel free to ask in this topic. We hope you enjoy our contest!
This post has been edited 3 times. Last edited by talkon, Mar 8, 2019, 3:21 PM
Reason: update
Reason: update
), please do so by 
are scores for submitted problems.) The total and average score for each problem are also provided.
