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University Problem Solving Thread

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Ahmad:
The purpose of this thread is to generate a lively discussion of mathematics and for FSNers to learn from each other by solving challenging problems!

Anyone can post a problem, but they have to be somewhat problem-solving type problems i.e. not the usual homework problem. I'm not sure about restricting what sort of problems you can pose, for example, posing something that uses sophisticated maths, but for now anything goes.

As the problems get posted I'll edit this post and add them in.  :)

1. Can you tile an 18 x 18 board (i.e. one divided into 324 squares) using only T-shaped pieces each made of 4 squares?

2. Prove that for reals .

3. A bug crawls along the edges of a cube; at each vertex it has probability of going to any adjacent vertex. When it reaches the vertex opposite its starting one, enlightenment is achieved. What is the average number of edges it must crawl on to do this?

4. Any right triangle contains an isosceles triangle whose area is at least times the area of the original triangle. Find the maximum value for .

5. Find

6. A certain Kingdom of Neobeoland introduces the following family-planning scheme: A couple will keep bearing children until a daughter is born, after which they will stop having children. Assume that they have an equal chance of birthing either gender. What will be the ratio of sons to daughters in this kingdom?

7. Find the smallest prime number which is a factor of 99! - 1.

Ahmad:
Problems added, source: Paradox. :)

Neobeo:
BUMP'd

I solved question 1 to be "yes". Now to draw a to-scale diagram.

dcc:
Question 6:

First look at the first few cases to get an idea for this:







So we want to find , that is, the mean number of males that we will have in Neobeoland:



which we can write as:



now rewriting this sum differently (THANKS AHMAD :)):



now the summation on the right is a variation of infinite geometric sequence, so we can rewrite as:





therefore if the expected number of males is 1, and there will ALWAYS be 1 female, so the ratio of males to females should be 1!

Neobeo:
Objection! Let's take a look at the problem from another angle. Instead of letting n be number of males, we let it be the proportion of females out of both genders. Note that I used dcc's template since I was too lazy to come up with my own.

Question 6:

First look at the first few cases to get an idea for this:









So we want to find , that is, the overall proportion of females that we will have in Neobeoland:



which we can write as:



now solving this sum directly (THANKS MATHEMATICA :)):



now the proportion of males of the whole population will just be:



Thus simplifying the ratio of sons to daughters to be:

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