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2018-03-05 Intermediate

         

Two points are placed randomly along the circumference of a circle. What is the probability that the chord drawn between them is longer than the radius of the circle?

How often is Length of Chord > Radius? How often is Length of Chord > Radius?

A positive integer nn has dd positive divisors.

After doubling n,n, you find that 2n2n has 2d2d positive divisors.

Is it possible that after tripling n,n, you will find that 3n3n has 3d3d 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 mm of the entire Earth's atmosphere?

Details and Assumptions:

  • The circumference of the Earth is about 40,000 km.40,000 \text{ km}.
  • Gravitational acceleration is approximately 10 m/s2.10 \text{ m/s}^2.
  • The mean air pressure on the Earth's surface is about 1 bar (1 bar=105 Pa).1 \text{ bar } \big(\SI{1}{bar} = \SI{10^5}{Pa} \big).

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

Then define the star power operation as an=aaaaawrittenntimes,a^{\star n}=\underbrace{a\star a\star a\star\cdots\star a}_{a\, \mathrm{written} \, n \, \mathrm{times}}, with a1=a.a^{\star 1}=a.

What is 77?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 360360^\circ of space surrounding the vertex without overlapping.

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

An equilateral triangle, regular 9-gon, and regular 18-gon can cover all \(360^\circ\) around a single vertex.  Is 18 the maximum number of sides one of these regular polygons can have? An equilateral triangle, regular 9-gon, and regular 18-gon can cover all 360360^\circ around a single vertex. Is 18 the maximum number of sides one of these regular polygons can have?

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