[/quote]
,
and 
is closest to which integer?
be a quadrilateral inscribed in a circle 
.
and 
meet at 
,
and 
meet at 
,
and 
meet at 
.
be the midpoint of the segment 
,
be the common point of the segment 
and the circle 
and 
be an acute triangle with orthocenter 
,
be a point on the side 
and 
.
are the feet of the altitudes from 
is the circumcircle of 
,
be the point on 
is a diameter of 
the circumcircle of triangle 
,
be the point such that 
is a diameter of 
and 
be an acute triangle. Define 
as its incenter. Let 
and 
be the incircle's tangent points to 
and 
,
be the midpoint of 
.
be the intersection point of a perpendicular line passing through 
.
intersects the circumcircle of 
at 
.
intersects line 
at 
.
is cyclic.
.![\[\angle PBC=\angle DBA\quad\text{and}\quad \angle PDC=\angle BDA.\]](http://latex.artofproblemsolving.com/c/2/0/c20761f3eadd054958f40259f3d1c05f26279783.png)
Prove that 
.
,
.
and 
.
lie on the same circle.
was chosen such that 
and 
.
.
,
.
and the external angle bisector of 
and 
,
.
and 
intersect at two points 
and 
.
be the line tangent to these circles at 
and 
,
be the line parallel to 
on 
on 
and 
meet at 
;
and 
;
and 
.
.
,
at 
respectively. Prove that 
.
Y by OlympusHero, Adventure10, Mango247, and 2 other users
I'm mainly preparing to score very high on the AIME, and I'm at the stage where I'm doing lots of practice tests. However, no matter how infrequently I do one, it is clear to me that I will run out of tests at some point. So I began wondering if there are competitions in other countries or earlier American competitions that would be comparable to the AIME and would help me prepare.
So, does anyone know of some competitions from other countries or earlier competitions that would help with AIME?
In particular, how does the Canadian Math Olympiad (CMO) compare with the AMC series here?
And are the USAMOs/TSTs/IMOs from a long time ago easier enough so that they are comparable to the modern AIME?
Thank you,
AwesomeToad
So, does anyone know of some competitions from other countries or earlier competitions that would help with AIME?
In particular, how does the Canadian Math Olympiad (CMO) compare with the AMC series here?
And are the USAMOs/TSTs/IMOs from a long time ago easier enough so that they are comparable to the modern AIME?
Thank you,
AwesomeToad
and wrote both the CMO and AIME this year. In preparing for these contests, I found that there was at least one CMO problem on every test that was either somewhat easy/trivial or which could be bashed using simple methods unlike some of the hardest problems on the AIME. I would say that the easiest two problems on the CMO tend to be between AIME 9 and 12 in difficulty. Problems 3 and 4 on the CMO are similar to the easiest problems on the USAMO, while the last problem can even as difficult as some USAMO #2s and #5s consider CMO 2008 #5 and 2007 #5).
