[/quote]
,
,
,
,
,
,
are shown in the figure below (not drawn to scale).The distances in kilometers along each route are marked. Traveling along these routes, what is the shortest distance from A to Z in kilometers?
chips, numbered 
,
,
,
,
for which there exists a sequence of distinct positive integers 
,
,
,
satisfying ![\[s_1^{s_2}=s_2^{s_3}=\cdots=s_{n-1}^{s_n}.\]](http://latex.artofproblemsolving.com/a/f/5/af594ff07aa2f89cb70ae669263558049556cf0e.png)
be an integer and 
is a sequence of real numbers for which 
and:
Prove that all elements of the sequence are positive.
be a positive integer, and let 
.
parts is a representation of n as a sum 
,
belong to 
.
parts with 
.![$\sqrt[3]{6n}$](http://latex.artofproblemsolving.com/2/0/1/201489b1f95ad200036698a606f712da7bb465b9.png)
different parts.
other knights where friendship is mutual. Prove that King Arthur can place some of his knights around a round table in such a way that every knight is friends with the 
friendships of knights that are not adjacent to each other.
is an integer. Some pairs of cities are connected by direct (two-way) flights. For two cities 
we define:
A 
between 
,
,
and 
for every 
;
A 
between 
A 
between 
be the total number of pairs of cities in the country that are connected by direct flights. In terms of 
,
.
.
are fixed positive integers.
with the following properties:
the following equality holds: 
such that
,
.
such that:
,
bisects neither the angle 
nor the angle 
.
lies inside ![\[\angle PBC=\angle DBA\quad\text{and}\quad \angle PDC=\angle BDA.\]](http://latex.artofproblemsolving.com/c/2/0/c20761f3eadd054958f40259f3d1c05f26279783.png)
Prove that 
.
and 
meet at a point 
be the point of the line 
,
.
and 
.
lie on the same circle.
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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.
