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2018-08-13 Intermediate

         

An ant named Honey is kept outside a solid cube on a leash, which extends to twice the side length of the cube.

The leash can be attached

  • on the center of a face, or
  • on the midpoint of an edge, or
  • on a vertex.

Which choice gives Honey the greatest area to roam?

\[\begin{align} 1 - \dfrac{1}{3} + \dfrac{1}{5} - \dfrac{1}{7} + \dfrac{1}{9} - \dfrac{1}{11} + \cdots & = \dfrac{\pi}{4} \\\\ \dfrac{1}{1 \times 3} + \dfrac{1}{5 \times 7} + \dfrac{1}{9 \times 11} + \cdots &=\dfrac{\pi}{\color{blue}x} \end{align}\]

What is \({\color{blue}x}?\)

The moment of inertia of a semi-circular wire about the diameter axis \(AB\) is \(I.\)

What is the moment of inertia about the axis \(CD\) passing through the center of the wire?

Hint: The moment of inertia depends on the mass and the distribution about the axis.

The triangle in this figure has side lengths 3, 4, 5. The arcs within the circle are semicircles.

What is the total area of the regions in blue?

Mr. Perfect can mow his lawn efficiently, that is, he can mow his whole lawn in a continuous line from start to finish without having to go over an already mown part, as shown by the blue path below.

No matter how hard he tries, Mr. Mediocre cannot mow his lawn (shown below) as efficiently as Mr. Perfect can. A friend suggests that Mr. Mediocre plant a shrub somewhere to change the layout of his yard.

Where should Mr. Mediocre plant the shrub so that he can mow his lawn efficiently?

Assumptions:

  • The lawnmower can only move straight through a square or make one right angle in the square.
  • The lawnmower cannot move diagonally between squares.
  • After the lawnmower enters and exits a square, all of the grass in the square is cut.

Bonus: Prove that before he plants the shrub, it is impossible for Mr. Mediocre to mow his lawn efficiently.

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