2022 AMC 8 Problems/Problem 1

Problem

The Math Team designed a logo shaped like a multiplication symbol, shown below on a grid of 1-inch squares. What is the area of the logo in square inches?

usepackage("mathptmx");
defaultpen(linewidth(0.5));
size(5cm);
defaultpen(fontsize(14pt));
label("$\textbf{Math}$", (2.1,3.7)--(3.9,3.7));
label("$\textbf{Team}$", (2.1,3)--(3.9,3));
filldraw((1,2)--(2,1)--(3,2)--(4,1)--(5,2)--(4,3)--(5,4)--(4,5)--(3,4)--(2,5)--(1,4)--(2,3)--(1,2)--cycle, mediumgray*0.5 + lightgray*0.5);

draw((0,0)--(6,0), gray);
draw((0,1)--(6,1), gray);
draw((0,2)--(6,2), gray);
draw((0,3)--(6,3), gray);
draw((0,4)--(6,4), gray);
draw((0,5)--(6,5), gray);
draw((0,6)--(6,6), gray);

draw((0,0)--(0,6), gray);
draw((1,0)--(1,6), gray);
draw((2,0)--(2,6), gray);
draw((3,0)--(3,6), gray);
draw((4,0)--(4,6), gray);
draw((5,0)--(5,6), gray);
draw((6,0)--(6,6), gray);
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Solution 1

Draw the following four lines as shown:

We see these lines split the figure into five squares with side length . Thus, the area is .

~pog ~wamofan

Solution 2

We can apply Pick's Theorem: There are lattice points in the interior and lattice points on the boundary of the figure. As a result, the area is .

~MathFun1000

Solution 3

Notice that the area of the figure is equal to the area of the square subtracted by the 12 triangles that are half the area of each square, which is 1. The total area of the triangles not in the figure is , so the answer is

~hh99754539

See Also

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.