2001 USAMO Problems/Problem 1

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Problem

Each of eight boxes contains six balls. Each ball has been colored with one of $n$ colors, such that no two balls in the same box are the same color, and no two colors occur together in more than one box. Determine, with justification, the smallest integer $n$ for which this is possible.

Solution

We claim that $n=23$ is the minimum. Consider the following construction (replacing colors with numbers) which fulfills this:

\[\begin{tabular}{|r|r|r|r|r|r|} \hline 1 & 2 & 3 & 4 & 5 & 6 \\ \hline 1 & 7 & 8 & 9 & 10 & 11 \\ \hline 1 & 12 & 13 & 14 & 15 & 16 \\ \hline 2 & 7 & 12 & 17 & 18 & 19 \\ \hline 3 & 8 & 13 & 17 & 20 & 21 \\ \hline 4 & 9 & 14 & 17 & 22 & 23 \\ \hline 5 & 10 & 15 & 18 & 20 & 22 \\ \hline 6 & 11 & 16 & 19 & 21 & 23 \\ \hline \end{tabular}\]

Suppose a configuration exists with $n \le 22$.

Suppose an element appears $5$ or more times. Then the remaining elements of the $5$ boxes must be distinct, so that there are at least $n \ge 5 \cdot 5 + 1 = 26$ elements, contradiction. If an element appears $4$ or more times, the remaining elements of the $4$ boxes must be distinct, leading to $5 \cdot 4 + 1 = 21$ elements. The fifth box can contain at most four elements from the previous boxes, and then the remaining two elements must be distinct, leading to $n \ge 2 + 21 = 23$, contradiction.

However, by the Pigeonhole Principle, at least one element must appear $3$ times. Without loss of generality suppose that $1$ appears three times, and let the boxes that contain these have the elements $\{1,2,3,4,5,6\},\{1,7,8,9,10,11\},\{1,12,13,14,15,16\}$. Each of the remaining five boxes can have at most $3$ elements from each of these boxes. Thus, each of the remaining five boxes must have $3$ additional elements $> 16$. Thus, it is necessary that we use $\le 22 - 16 = 6$ balls to fill a $3 \times 5$ grid by the same rules.

Again, no element may appear $\ge 4$ times, but by Pigeonhole, one element must appear $3$ times. Without loss of generality, let this element be $17$; then the three boxes containing $17$ must have at least $2 \cdot 3 + 1 = 7$ elements, contradiction.

Therefore, $n = 23$ is the minimum.

Alternate Solution:

For each box, there are 6C2pairs of colors. Since no two colors can appear in more than one box together, the 8*6C2 pairs must be distinct. With x being the total number of colors, if xC2>8*6C2 is satisfied, the x colors can be arranged in a manner that satisfies the conditions.

Solve, and x=23 follows as the least number of colors possible that satisfies the conditions.

See also

2001 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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