% © Evan Chen
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\title{USAMO 2011 Solution Notes}
\date{\today}
\begin{document}
\maketitle
\begin{abstract}
This is a compilation of solutions
for the 2011 USAMO.
The ideas of the solution are a mix of my own work,
the solutions provided by the competition organizers,
and solutions found by the community.
However, all the writing is maintained by me.
These notes will tend to be a bit more advanced and terse than the ``official''
solutions from the organizers.
In particular, if a theorem or technique is not known to beginners
but is still considered ``standard'', then I often prefer to
use this theory anyways, rather than try to work around or conceal it.
For example, in geometry problems I typically use directed angles
without further comment, rather than awkwardly work around configuration issues.
Similarly, sentences like ``let $\mathbb{R}$ denote the set of real numbers''
are typically omitted entirely.
Corrections and comments are welcome!
\end{abstract}
\tableofcontents
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\section{Problems}
\begin{enumerate}[\bfseries 1.]
\item %% Problem 1
Let $a$, $b$, $c$ be positive real numbers
such that $a^2+b^2+c^2+(a+b+c)^2 \le 4$. Prove that
\[ \frac{ab+1}{(a+b)^2}
+ \frac{bc+1}{(b+c)^2}
+ \frac{ca+1}{(c+a)^2} \ge 3. \]
\item %% Problem 2
An integer is assigned to each vertex of a regular pentagon
so that the sum of the five integers is $2011$.
A turn of a solitaire game consists of subtracting an integer $m$
(not necessarily positive) from each of the integers at two neighboring vertices
and adding $2m$ to the opposite vertex, which is not adjacent
to either of the first two vertices.
(The amount $m$ and the vertices chosen can vary from turn to turn.)
The game is won at a certain vertex if, after some number of turns,
that vertex has the number $2011$ and the other four vertices have the number $0$.
Prove that for any choice of the initial integers,
there is exactly one vertex at which the game can be won.
\item %% Problem 3
In hexagon $ABCDEF$, which is nonconvex but not self-intersecting,
no pair of opposite sides are parallel.
The internal angles satisfy
$\angle A=3\angle D$, $\angle C=3\angle F$, and $\angle E=3\angle B$.
Furthermore $AB=DE$, $BC=EF$, and $CD=FA$.
Prove that diagonals $\ol{AD}$, $\ol{BE}$, and $\ol{CF}$
are concurrent.
\item %% Problem 4
Consider the assertion that for each positive integer $n\geq2$,
the remainder upon dividing $2^{2^n}$ by $2^n-1$ is a power of $4$.
Either prove the assertion or find (with proof) a counterexample.
\item %% Problem 5
Let $P$ be a point inside convex quadrilateral $ABCD$.
Points $Q_1$ and $Q_2$ are located within $ABCD$ such that
\begin{align*}
\angle Q_1BC=\angle ABP, & \qquad \angle Q_1CB=\angle DCP, \\
\angle Q_2AD=\angle BAP, & \qquad \angle Q_2DA=\angle CDP.
\end{align*}
Prove that $\ol{Q_1Q_2} \parallel \ol{AB}$
if and only if $\ol{Q_1Q_2} \parallel \ol{CD}$.
\item %% Problem 6
Let $A$ be a set with $|A|=225$, meaning that $A$ has $225$ elements.
Suppose further that there are eleven subsets $A_1, \dots, A_{11}$ of $A$
such that $|A_i|=45$ for $1\leq i\leq11$ and $|A_i\cap A_j|=9$
for $1\leq i