Maths Olympiad Prep

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Geometry Difficulty 5.8 AIME, harder Prove it United States

Problem:

Sasha wants to bake 6 cookies in his 88 inch ×\times 88 inch square baking sheet. With a cookie cutter, he cuts out from the dough six circular shapes, each exactly 33 inches in diameter. Can he place these six dough shapes on the baking sheet without the shapes touching each other? If yes, show us how. If no, explain why not. (Assume that the dough does not expand during baking.)

Solution

Solution:

Yes, but just barely! The centers of the cookies must lie within the middle 5×55 \times 5 square. Divide this square in half one way and in thirds the other way, creating a grid of six 52×53\frac{5}{2} \times \frac{5}{3} rectangles, and place cookie centers at alternate intersections of this grid as shown.

Figure 1

Then the distance between nearest neighboring cookie centers is
(52)2+(53)2=3256>3 \sqrt{\left(\frac{5}{2}\right)^{2}+\left(\frac{5}{3}\right)^{2}}=\frac{\sqrt{325}}{6}>3
so the cookies do not touch or overlap.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.