2007 USAMO Problems/Problem 4

Problem

An animal with $n$ cells is a connected figure consisting of $n$ equal-sized cells. ${}^1$ The figure below shows an 8-cell animal.

(insert picture here)

A dinosaur is an animal with at least 2007 cells. It is said to be primitive it its cells cannot be partitioned into two or more dinosaurs. Find with proof the maximum number of cells in a primitive dinosaur.

${}^1$ Animals are also called polyominoes. They can be defined inductively. Two cells are adjacent if they share a complete edge. A single cell is an animal, and given an animal with $n$ cells, one with $n+1$ cells is obtained by adjoining a new cell by making it adjacent to one or more existing cells.

Solution

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Note that for the dinosaur to have $\ge$ 4014 cells then the answer is $4(2007-1)+1=8025$ .

2007 USAMO (Problems • Resources)
Preceded by Problem 3	Followed by Problem 5
1 • 2 • 3 • 4 • 5 • 6
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2007 USAMO Problems/Problem 4

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