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2018-11-12 Intermediate

         

One hundred boxes are in a row, and a precious diamond is hidden inside one of them. You don't know which box has the diamond.

All the boxes are labeled the same, as shown below, but only one of these boxes is telling the truth.

What is the minimum number of boxes you need to open to be sure about which box contains the diamond?

The highest intensity of sunlight is only available on Earth’s equator. At any other latitude, the incident sunlight is spread over a larger area, decreasing the intensity of the sunlight.

The amount of solar energy collected by a single solar panel can be maximized by tilting the panel. For example, to collect the greatest amount of energy with a single solar panel in Greece at \(40^{\circ} \textrm{N}\) latitude, you should angle the panel at \(40^{\circ}\).

Suppose you are building a solar farm on level ground to power a town in Greece. To decrease your environmental impact, you would like to minimize the square-shaped land area affected by the the solar farm – no one wants a row of unsightly panels around the town or a shadow cast on their backyard. Your solar farm needs employ as many commercially available solar panels as necessary to power the nearby town.

How much land area can be saved by tilting the solar panels at \(40^{\circ}\) with one edge on the ground vs. laying them flat?

This barrel filled with water is located on the ground and has 7 equally spaced, vertically aligned holes. Hole number four is exactly at the middle of the barrel.

Assuming that there is no air resistance, viscosity, or other hindrance to the water flow, which hole will jet the water farther than any other holes do?

When a conical bottle rests on its flat base, the water in the bottle is 8 cm from the cone's vertex.

When the same conical bottle is turned upside down, the water surface is 2 cm from the flat base.

What is the height (in cm) of the bottle?

Eight pieces of string lie side by side on a table.

You tie the top ends of the strings together into four random pairs, and then do the same for the bottom ends.

What is the probability (as a decimal) that all of the strings have been tied into one single loop?

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