\section{Extra Problems}


\problem{USAMO 1996 Problem 6}
Determine (with proof) whether there is a subset $X$ of
the nonnegative integers with the following property: for any nonnegative integer $n$ there is exactly
one solution of $a + 2b = n$ with $a, b \in X$.
(The original USAMO question asked about all integers, not just nonnegative - this is harder,
but still approachable with generating functions.)


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\problem{IMO Shortlist 1998}
Let $a_0, a_1, ...$ be an increasing sequence of nonnegative integers
such that every nonnegative integer can be
expressed uniquely in the form $a_i + 2a_j + 4a_k$,
where $i, j, k$ are not necessarily distinct.

Determine $a_1998$.


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\problem{USAMO 1986 Problem 5}
By a partition $\pi$ of an integer $n \geq 1$, we mean here a
representation of $n$ as a sum of one or more positive integers where the summands must be put in
nondecreasing order. (e.g., if $n = 4$, then the partitions $\pi$ are
$1 + 1 + 1 + 1$, $1 + 1 + 2$, $1 + 3, 2 + 2$, and $4$).


For any partition $\pi$, define $A(\pi)$ to be the number of ones which appear in $\pi$, and define $B(\pi)$
to be the number of distinct integers which appear in $\pi$ (e.g, if $n = 13$ and $\pi$ is the partition
$1 + 1 + 2 + 2 + 2 + 5$, then $A(\pi) = 2$ and $B(\pi) = 3$).

Show that for any fixed $n$, the sum of $A(\pi)$ over all partitions of $\pi$ of $n$ is equal to the sum of
$B(\pi)$ over all partitions of $\pi$ of $n$.

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\problem{USAMO 2017 Problem 2}
Let $m_1, m_2, ..., m_n$ be a collection of $n$ distinct positive
integers. For any sequence of integers $A = (a_1, ..., a_n)$ and any permutation $w = w_1, ..., w_n$ of
$m_1, ..., m_n$, define an $A$-inversion of $w$ to be a pair of entries $w_i, w_j$ with $i < j$ for which one of the
following conditions holds:
\begin{itemize}
	\item $ai \geq wi > wj$
	\item $wj > ai \geq wi$
	\item $wi > wj > ai$
\end{itemize}

Show that for any two sequences of integers $A = (a_1, ..., a_n)$ and $B = (b_1, ..., b_n)$ and for any
positive integer $k$, the number of permutations of $m_1, ..., m_n$ having exactly $k$ $A$-inversions is equal
to the number of permutations of $m_1, ..., m_n$ having exactly $k$ $B$-inversions.
(The original USAMO problem allowed the numbers $m_1, ..., m_n$ to not necessarily be distinct.)

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