[/quote]
,
such that 
coprime, 
and 
.
be a nonempty subset of the points in the Cartesian plane such that for each 
exactly one of 
or 
also belongs to 
there is a line in the plane (possibly different lines for different 
be a function such that for all positive rational numbers 
,![\[
f\left(\frac{f(x)}{y f(x) + 1}\right) = \frac{x}{x f(y) + 1}.
\]](http://latex.artofproblemsolving.com/e/e/a/eea34789c7b2fa698ddaff29d95f4af0a71850b5.png)
and 
,
to be the same color. The directed-edge-chromatic-number of a tournament is defined to be the minimum total number of colors that can be used in order to create a proper directed-edge-coloring. For each 
,
Prove that
Let 
Prove that![$$ (a+b)(a+1)(b+1) \leq \frac{3}{2}+\sqrt[3]{6}+\sqrt[3]{36}$$](http://latex.artofproblemsolving.com/f/5/e/f5e68204507aa9c3e71c28844bbee3a804546b94.png)
be an acute triangle. Let 
be the altitude on 
,
be any interior point on 
,
at 
respectively. Prove that 
.
is the midpoint of 
is a point on the side 
.
,
,
are positive integers 
.
is also an integer and equals 
.
,
and 
.
and 
intersect the circumcircle of 
and 
,
and 
and 
are parallelograms (with 
,
)
be the point of intersection of lines 
and 
.
,
and a positive integer 
,
is a perfect square.
with respect to 
is greater than 0 if 
and 
.
and since 
?
Y by Adventure10, Mango247
OK, well, I waste a lot of time, and so I don't think I'm time-efficient enough to do WOOT this year, sadly. However, my last 3 practice AIME scores have been 7,5, and 9, and I feel I can consistently score 7-9 (the 5 was late at night, and was very computational, and had lots of geo, one of which is good for my AIME performance. Luckily I take AIME for real in the morning/afternoon) and I feel I am getting to the point (after a few more practice AIMEs) where I feel I can start transitioning to Olympiad problems. However, I feel this will be hard without WOOT, but I may be able to take Olympiad Geo. Will going through practice tests really work? I plan to go from early USAMO's and CMO's to harder contests and problems, but olympiad problems are hard. Luckily, I have solved one (the JMO #1 in practice, does that count?) but I feel that practice tests alone won't work (well, they did for AIME, with a great MathPath breakout course, but I don't know). My ultimate goal next year is to win JMO. I want to find a book/books or other ways to study that give me a fighting chance. USAJMO problems, I feel, are at or a little above my level. I've heard ACOPS and PSS are good, but I can't stand how ACOPS doesn't have solutions. I think I'll go for these books (ACOPS and PSS) nonetheless, but does anyone have any suggestions?
EDIT: Also, I'm OK at, but not very comfortable with logs, preventing me from doing the last step of (spoilers) and for trig, I know sum identities, law of cosines and sines, sort of extended law of signs, but no product identities or stuff like that. How should I learn these so as not to be defeated by a trig or log question on JMO?
Thank you for reading through this long post and thanks in advance for advice,
-Yugrey
EDIT: Also, I'm OK at, but not very comfortable with logs, preventing me from doing the last step of (spoilers) and for trig, I know sum identities, law of cosines and sines, sort of extended law of signs, but no product identities or stuff like that. How should I learn these so as not to be defeated by a trig or log question on JMO?
Thank you for reading through this long post and thanks in advance for advice,
-Yugrey
.
Should I just keep working through early USAMOs? It turns out there is a small chance I get to take WOOt, not a 0 chance! 
(or just replace the 176 with 180).
.
