[/quote]
,
such that 
is an integer for every positive integer 
be a triangle, and let 
be a point on side 
.
at 
and ray 
at 
. The circumcircle of triangle 
intersects the circumcircle 
of triangle 
distinct from point 
.
and 
intersect 
and 
respectively.
grid has had the central unit square removed. For which positive integers 
is it possible to cut the remaining area into 
and 
rectangles?
be positive real numbers so that 
.![\[ \left( a - 1 + \frac 1b \right) \left( b - 1 + \frac 1c \right) \left( c - 1 + \frac 1a \right) \leq 1.
\]](http://latex.artofproblemsolving.com/7/6/e/76e794a62090c8c4b5f3a70ad4d036669418ca69.png)
and 
be positive real numbers such that 
.![\[
\frac{x^{3}}{(1 + y)(1 + z)}+\frac{y^{3}}{(1 + z)(1 + x)}+\frac{z^{3}}{(1 + x)(1 + y)}
\geq \frac{3}{4}.
\]](http://latex.artofproblemsolving.com/5/6/5/56568d27a296f1c900fabbb1042549cd2e552e29.png)
,![\[ \frac{a}{\sqrt{a^2 + 8bc}} + \frac{b}{\sqrt{b^2 + 8ca}} + \frac{c}{\sqrt{c^2 + 8ab}} \geq 1. \]](http://latex.artofproblemsolving.com/b/2/f/b2ff456632d4dd3b9ed93e4cba437bdd34d54cd8.png)
be a parallelogram of center 
.
,
and 
such that 
.
be given . On the extension of the segment 
beyond 
is chosen arbitrarily. Let the line through 
intersect the line 
.
be the intersection of the lines 
and 
.
is tangent to the circumcircle of the triangle 
.
be any point. Let 
be projection point from 
.
cuts circumcircle 
at 
.
be antipode of 
are cyclic.
Y by Pengu14, Alex-131
hey!! im a first time applicant for a lot of math camps (namely: usa-canada mathcamp, PROMYS, Ross, MathILY, HCSSiM), and I was just wondering:
1. how much of an effect would being a first-time applicant have on making these math camps individually?
2. I spent a huge amount of effort (like 50 or something hours) on the USA-Canada Mathcamp application quiz in particular, but I'm pretty worried because supposedly almost no first-time applicants get into the camp. Are there any first-time applicants that you know of, and what did their applications (as in, qualifying quiz solutions) look like?
3. Additionally, a lot of people give off the impression that not doing the full problem set will screw your application over, except in rare cases. How much do you think a fakesolve would impact my PROMYS application chances?
thanks in advance!
1. how much of an effect would being a first-time applicant have on making these math camps individually?
2. I spent a huge amount of effort (like 50 or something hours) on the USA-Canada Mathcamp application quiz in particular, but I'm pretty worried because supposedly almost no first-time applicants get into the camp. Are there any first-time applicants that you know of, and what did their applications (as in, qualifying quiz solutions) look like?
3. Additionally, a lot of people give off the impression that not doing the full problem set will screw your application over, except in rare cases. How much do you think a fakesolve would impact my PROMYS application chances?
thanks in advance!
)
