[/quote]
,
and 
be the endpoints of a semicircular arc of radius 
.
.
or 
are drawn. Let 
be the product of the lengths of these twelve chords. Find the remainder when 
.
be non-negative real numbers such that 
Prove that
and 
.
,![\[ a^{b+c} b^{c+a} c^{a+b} \leq 1 . \]](http://latex.artofproblemsolving.com/0/c/e/0ce097b0b507403c959c2992cae1217ecdd7e581.png)
be a finite sequence of positive integers. Prove that there exist nonnegative integers 
and 
such that 
holds for all integers 
be a positive integer. Dominoes are placed on a 
board in such a way that every cell of the board is adjacent to exactly one cell covered by a domino. For each 
or 
.
of positive real numbers that satisfy the system:![\[
\begin{aligned}
11bc - 36b - 15c &= abc \\
12ca - 10c - 28a &= abc \\
13ab - 21a - 6b &= abc.
\end{aligned}
\]](http://latex.artofproblemsolving.com/1/7/0/170e1cfaee90f5c37428fa75192ac889261a8f83.png)
be positive reals. Prove that there is a cyclic permutation 
of 
such that the inequality:![\[
\frac{a}{xa + yb + zc} + \frac{b}{xb + yc + za} + \frac{c}{xc + ya + zb} \geq \frac{3}{x + y + z}
\]](http://latex.artofproblemsolving.com/5/5/3/553527fc40a1b7d29dbdfb4b5e63635308f46c85.png)
and 
.
be a scalene triangle with orthocenter 
and circumcenter 
.
,
,
.
with diameter 
,
at a point 
.
meets line 
at 
.
and 
intersect at a point 
on 
.
Prove that: 
is the sum of 
square numbers
Let 
and 
is the sum of 
is the sum of 
Y by aidan0626, Quidditch
Hello AOPS Community,
Recently, me and my friend compiled a set of the most high quality problems from our imagination into a problem set called the Million. This series has three contests, called the whun, thousand and Million respectively.
Unfortunately, it did not get the love it deserved on the OTIS discord. Hence, we post it here to share these problems with the AOPS community and hopefully allow all of you to enjoy these very interesting problems.
Good luck! Lastly, remember that MILLION ORZ!
Edit: We have just been informed this will be the Orange MOP series. Please pay close attention to these problems!
Recently, me and my friend compiled a set of the most high quality problems from our imagination into a problem set called the Million. This series has three contests, called the whun, thousand and Million respectively.
Unfortunately, it did not get the love it deserved on the OTIS discord. Hence, we post it here to share these problems with the AOPS community and hopefully allow all of you to enjoy these very interesting problems.
Good luck! Lastly, remember that MILLION ORZ!
Edit: We have just been informed this will be the Orange MOP series. Please pay close attention to these problems!
This post has been edited 2 times. Last edited by Mathdreams, Apr 2, 2025, 3:59 AM
