A positive integer \(n\) has \(d\) positive divisors.

After doubling \(n,\) you find that \(2n\) has \(2d\) positive divisors.

Is it possible that after tripling \(n,\) you will find that \(3n\) has \(3d\) positive divisors?

The atmosphere is a layer of gas that surrounds our planet and is several hundred kilometers thick. Its main components are nitrogen, oxygen, argon, and water vapor.

What is the approximate mass \(m\) of the entire Earth's atmosphere?

**Details and Assumptions:**

- The circumference of the Earth is about \(40,000 \text{ km}.\)
- Gravitational acceleration is approximately \(10 \text{ m/s}^2.\)
- The mean air pressure on the Earth's surface is about \(1 \text{ bar } \big(\SI{1}{bar} = \SI{10^5}{Pa} \big).\)

Define the **star** operation as \(a\star b=a+ab+b.\)

Then define the **star power** operation as \(a^{\star n}=\underbrace{a\star a\star a\star\cdots\star a}_{a\, \mathrm{written} \, n \, \mathrm{times}},\) with \(a^{\star 1}=a.\)

What is \(7^{\star7}?\)

An equilateral triangle and 2 other regular polygons share a single vertex in such a way that the three shapes completely cover the \(360^\circ\) of space surrounding the vertex without overlapping.

What is the largest possible number of sides that one of the regular polygons can have?

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