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TAMU-CC is participating in 'The Problem Solving Competition' as sponsored by Dr. Richard S. Neal of the ASCM.

Rules

1. The competition is open to all undergraduate students at TAMU-CC.
2. Problems, Rules and Solutions will be posted regularly on the bulletin board near Dr. Sterba-Boatwright's office. The problem also will be printed on smaller slips of paper to take away and solve later.
3. Deadlines will be posted for problems. Solutions are due under the door of CI 310 by 2 PM of the given deadline. Solutions should be dated and signed. The name of the solver and contact information (phone, email) should be clearly written on the paper.
4. Solutions will be judged according to correctness, timeliness, and then aesthetic measures. A solution must have a detailed explanation, rather than just say something like "224".
5. The student producing the best solution will be declared the winner and awarded a certificate and a prize. Students winning one month are not eligible to win the next month unless there are no other winners.
6. Cooperative solutions cannot be winners, but they are better than no solution at all.
7. All decisions of the Math Club president and advisor about the competition are final.
8. Have fun!

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September problems:

To be won: $5 for each problem, donated by Dr. Sterba-Boatwright

Problem 1: Let {$ g(x)= \ln(x)$}. Find

{$\lim_{p\to 0}\left(\lim_{r\to 0}\left( \frac{g(27+27p+9p^2+p^3)-g(27)}{4pr}-{{12g(3+p+r)-12g(3+r)}\over{16 pr}}\right)\right)$}. Show your work in detail.

Problem 2: Starting with 5 colors, say red, white, blue, yellow and green, how many ways can the corners (vertices) of a square be colored? Note that a vertex is dimensionless so it can't actually be "colored". It is more proper to say we are assigning colors to the vertices. Assume that reflections and rotations are allowed. (This means that if you examine a square from the front or back it represents the same coloring and if you turn it it represents the same coloring. Also assume that no vertex is left uncolored. Show and explain your work!

Deadline: October 6, 2006, 2 PM

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