Problem: An \emph{animal} with $n$ \emph{cells}
is a connected figure consisting of $n$ equal-sized square cells
(equivalently, a polyomino with $n$ cells).
A \emph{dinosaur} is an animal with at least $2007$ cells.
It is said to be \emph{primitive}
it its cells cannot be partitioned into two or more dinosaurs.
Find with proof the maximum number of cells in a primitive dinosaur.

Solution: In fact it's true for any tree with maximum degree $\le 4$.
Here is the solution of Andrew Geng.

Let $T$ be such a tree (a spanning tree of the dinosaur graph)
which corresponds to a primitive dinosaur.

\begin{claim*}
  There exists a vertex $v$ such that when $v$ is deleted, no dinosaurs result.
\end{claim*}
\begin{proof}
  Assume for contradiction that all vertices are bad
  (leave a dinosaur when deleted).
  Consider two adjacent vertices $v$, $w$ in $T$.
  By checking possibilities, one sees that, say,
  the dinosaur in $T-v$ contains $w$ and the dinosaur of $T-w$.
  We can repeat in this way; since $T$ is acyclic,
  this eventually becomes a contradiction.
\end{proof}

When this vertex is deleted,
we get at most $4$ components, each with $\le 2006$ vertices,
giving the answer of $4 \cdot 2006 + 1 = 8025$.
The construction is easy (take a ``cross'', for example).