2006 SMT/Advanced Topics Problems
Contents
[hide]Problem 1
A college student is about to break up with her boyfriend, a mathematics major who is apparently more interested in math than her. Frustrated, she cries, ”You mathematicians have no soul! It’s all numbers and equations! What is the root of your incompetence?!” Her boyfriend assumes she means the square root of himself, or the square root of . What two answers should he give?
Problem 2
Define . Find a vertical vector such that (where is the identity matrix).
Problem 3
Simplify: (Your answer should contain no summations but may still contain binomial coefficients/combinations).
Problem 4
Rice University and Stanford University write questions and corresponding solutions for a high school math tournament. The Rice group writes questions every hour but make a mistake in calculating their solutions of the time. The Stanford group writes problems every hour and makes solution mistakes of the time. Each school works for hours and then sends all problems to Smartie to be checked. However, Smartie isn’t really so smart, and only of the problems she thinks are wrong are actually incorrect. Smartie thinks of questions from Rice have incorrect solutions, and that of questions from Stanford have incorrect solutions. This problem was definitely written and solved correctly. What is the probability that Smartie thinks its solution is wrong?
Problem 5
Evaluate:
Problem 6
Ten teams of five runners each compete in a cross-country race. A runner finishing in place contributes points to his team, and there are no ties. The team with the lowest score wins. Assuming the first place team does not have the same score as any other team, how many winning scores are possible?
Problem 7
A lattice point in the plane is a point whose coordinates are both integers. Given a set of distinct lattice points in the plane, find the smallest number of line segments for which and are distinct lattice points in this set and the midpoint of is also a lattice point (not necessarily in the set).
Problem 8
The following computation arose in the research of mathematician P.D.: Let for .
Problem 9
How many positive integers appear in the list where represents the greatest integer that does not exceed ?
Problem 10
Evaluate: