1993 AMC 8 Problem 9

Below is the professionally curated solution for Problem 9 of the 1993 AMC 8, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1993 AMC 8 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

Difficulty rating: 730

Consider the operation $*$ defined by the following table:

$\begin{array}{c|cccc} * & 1 & 2 & 3 & 4 \\ \hline 1 & 1 & 2 & 3 & 4 \\ 2 & 2 & 4 & 1 & 3 \\ 3 & 3 & 1 & 4 & 2 \\ 4 & 4 & 3 & 2 & 1 \end{array}$

For example, $3 * 2 = 1.$ Then $(2 * 4) * (1 * 3) =$

$1$

$2$

$3$

$4$

$5$

From the table, $2 * 4 = 3$ and $1 * 3 = 3.$

Then $(2 * 4) * (1 * 3) = 3 * 3 = 4.$

Thus, the correct answer is D .