1999 USAMO Problems/Problem 1

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Some checkers placed on an $n\times n$ checkerboard satisfy the following conditions:

(a) every square that does not contain a checker shares a side with one that does;

(b) given any pair of squares that contain checkers, there is a sequence of squares containing checkers, starting and ending with the given squares, such that every two consecutive squares of the sequence share a side.

Prove that at least $(n^{2}-2)/3$ checkers have been placed on the board.

Solution 1

For the proof let's look at the checkers board as a graph $G$, where the checkers are the vertices and the edges are every pair of checkers sharing a side.

Define $R$ as the circuit rank of $G$(the minimum number of edges to remove from a graph to remove all its cycles).

Define $x$ as the number of checkers placed on the board.

1. From (a) for every square containing a checker there can be up to 4 distinct squares without checkers; thus, $4\times x + x = 5\times x \ge n^2$ (this is the most naive upper bound).

2. From (b) this graph has to be connected. which means that no square which contains a checker can share 4 sides with squares that doesn't contain a checker (because it has to share at least one side with a square that contains a checker).

3. Now it is possible to improve the upper bound. Since every edge in G represents a square with a checker that shares a side with a square with another checker, the new upper bound is $5\times x - \text{[sum of degrees in G]} \ge n^2.$

4. Thus, $5x - (2x - 2 + 2R) \ge n^2$ (see lemma below).

5. Thus, $3x + 2 - 2R \ge n^2.$

6. Thus, $x \ge \frac{n^2 - 2 + 2R}3,$ where $R\ge0$ and is equal to 0 if $G$ is a tree.

$\textit{Clarification/Lemma:}$ The sum of degrees of a connected graph $G = (V,E)$ is $2V -2 + 2R = 2E,$ where $R$ is the circuit rank of $G$.

$\textit{Proof.}$ $2 \times V -2$ is the sum of ranks of the spanning tree created by decircuiting the graph $G$. Since the circuit rank of $G$ is $R$, $2R$ is the sum of ranks removed from $G$. Thus, $2V -2 + 2R = 2 E.$ $\blacksquare$

Solution 2

Call a square of the checkerboard “good” if it either has a checker on it or shares a side with a square with a checker on it.

Say there are $k$ checkers on the board, sitting on squares $s_1, \ldots, s_k$. We can assume WLOG that $s_1, \ldots, s_k$ is ordered in such a way that if $1 \leq i \leq k$, then $s_i$ shares a side with at least one of $s_1, \ldots, s_{i - 1}$. (If such an ordering were impossible, then there would be an $s_i$ which isn’t connected to $s_1$ by a sequence of squares with checkers on them, violating condition (b).)

Now imagine removing all the checkers and then putting them back onto the squares $s_1, \ldots, s_k$ in that order, counting the number of good squares after each step. When we put a checker on $s_1$, we increase the number of good squares by at most 5: the square $s_1$ becomes good, and each square it shares a side with (at most 4) becomes good. When we put a checker onto $s_i$, where $i > 1$, we increase the number of good squares by at most $3$, since $s_i$ is already good, and at least $1$ of its at most $4$ neighbors already has a checker on it.

So the total number of good squares when we’ve put back every checker is at most $5 + 3(k - 1) = 3k + 2$. In order to satisfy condition (a), we have to make each of the $n^2$ squares good. In other words, we need $3k + 2 \geq n^2$, or \[k \geq \frac{n^2 - 2}{3}.\]

See Also

1999 USAMO (ProblemsResources)
Preceded by
First Question
Followed by
Problem 2
1 2 3 4 5 6
All USAMO Problems and Solutions

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