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2017-10-09 Basic

         

The equilateral triangle, circle, and square shown above have equal areas. Which has the largest perimeter?

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Note: The diagram is not drawn to scale.

I want to fill in the eight circles below using each of the numbers 1, 2, 3, 4, 5, 6, 7, 8 exactly once. Additionally, consecutive numbers--e.g. \((1, 2)\) or \((5,6)\)-- cannot be placed in circles which are connected by a line segment.

Is this possible, and if so, what is the sum of the two numbers in the middle?

A pendulum, which consists of a hollow cylinder filled with water, is set to swing. If the cylinder is punctured at the bottom so that water starts to flow out, what will happen to the period of the pendulum over time?

On an \(8 \times 8\) chessboard, can a knight move from one corner to the opposite corner, visiting each and every remaining square exactly once on its way there?

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Clarification: A knight can move to a square that is either two squares away horizontally and one square vertically, or two squares vertically and one square horizontally.

Hasmik invited several guests, and she knows that there will either be 7 or 8 people at the party. She wants to slice up a big pie into smaller pieces, not necessarily of the same size, such that regardless of how many people show up, she can serve the entire pie evenly to everyone.

What is the minimum number of pieces she will need to slice the big pie into?

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