[/quote]
,
be a positive integer, with divisors 
.
is always less than 
,
and 
be sequences of real numbers for which 
and
for all positive integers 
.
for which 
for all 
)
be an integer. Let 
,
,
be positive real numbers such that ![\[n^2 + 1 > \left( t_1 + t_2 + \cdots + t_n \right) \left( \frac{1}{t_1} + \frac{1}{t_2} + \cdots + \frac{1}{t_n} \right).\]](http://latex.artofproblemsolving.com/1/a/b/1ab410ebc17d9f6c38a3e7ae0f6efa955ffddfd3.png)
Show that 
,
,
are side lengths of a triangle for all 
,
,
with 
.
be an acute triangle with incenter 
and circumcenter 
.
and 
at 
and 
respectively, and 
is the reflection of 
over 
and 
meet at 
,
and 
,
is the midpoint of 
.
and 
,
.
and 
.![\[P=\frac{1}{5-2a}+\frac{1}{5-2b}+\frac{1}{5-2c}.\]](http://latex.artofproblemsolving.com/c/3/7/c3708768a84c2bbb1e4fae502a9bb1e92cb13d02.png)
and 
Prove that
but 
does not divide 
.
with right angles at 
has an inscribed circle with center 
.
and 
be 
and 
,
and 
be 
.
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An essay I wrote about my math experiences (including experiences with USAMO)
Please feel free to comment, positively or negatively, and share your own experiences.
However, I respectfully ask that you do not comment until you have read the entire essay.
Thank you.
Please feel free to comment, positively or negatively, and share your own experiences.
However, I respectfully ask that you do not comment until you have read the entire essay.
Thank you.
