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2017-03-13 Advanced


True, False, or Conditionally True?

We can cut any rectangle into three pieces using straight lines, and rearrange these pieces into a convex hexagon whose sides are all of the same length.

Let's make a cone out of a circular piece of paper.
We first cut out a sector, and connect the radius in the remaining sector, as demonstrated by the animation.

To maximize the volume of the resultant cone, what must be the central angle (in degrees, to 2 decimal places) of the remaining sector?

A pianist who owns a rare antique four-legged piano hires a professional piano moving company to have it moved into his studio. After they've moved the piano, the movers present him with a bill based on the weight of the piano, which they say is \(900\) pounds. When asked by the pianist how they came to that figure, they explain their procedure for grand pianos which usually have three legs:

1. Lift up one leg slightly and slide a weighing scale underneath.
2. Record down the weight on the scale.
3. Repeat this for all the other legs.
4. Sum up the weights.

The movers show him a diagram of the piano, and the weights recorded at each of the legs.

They conclude that this piano weighs \(200+200+200+300=900\) pounds.

The pianist objects, saying that there is no way that piano can weigh \(900\) pounds. He suspects that the back legs are double-counted, and insists that the piano only weighs \( 200 + 200 + 300 = 700 \) pounds.

As it turns out, neither of them are right. What is the true weight of the piano, in pounds?

Note: The scale is not paper-thin, it does lift the piano leg a little off the floor.

Find the maximum value (to 3 decimal places) of
\[\frac {ab + bc + cd}{a^2 + b^2 + c^2 + d^2}\] for reals \(a\), \(b\), \(c\), and \(d\) which are not all zero.

The city of Oneway has a \( 3 \times 5 \) park, and a \( 513 \times 1025 \) grassland that they own.

To make it easier to run search and rescue, they want to fill several grid squares with huge trees that prevent passage, so that in the remaining area there is exactly one path between any two grid squares that does not revisit any cells. (We only allow for moving between squares that share a side, and not squares that share a corner.)

For the \( 3 \times 5 \) park, several proposals were submitted, of which 2 are displayed:

The proposal on the left was immediately rejected because there were two ways to go between A and B, and no ways to go between A and C. The proposal on the right was accepted, as it fulfilled their conditions.

The town is under some economic hardship, so they will award the contract to the construction company that uses the fewest trees. For the park, they eventually awarded the contract to another proposal that only used 3 squares of trees.

The \( 513 \times 1025 \) grassland represents a huge contract that your company wants to be awarded. Determine the minimum number of squares of trees that are needed to fulfill the desires of the town, and allow your company to come out on top.


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