2006 SMT/Advanced Topics Problems

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 $i$ . What two answers should he give?

Solution

Problem 2

Define $A=\left(\,\,\begin{matrix}0 & 1\\ 3 &0\end{small}\right)$ (Error compiling LaTeX. Unknown error_msg). Find a vertical vector $v$ such that $(A^8+A^6+A^4+A^2+I)v=\left(\begin{matrix}0\\11\end{matrix}\right)$ (where $I$ is the $2\times2$ identity matrix).

Solution

Problem 3

Simplify: $\sum_{k=10}^{2006}\binom{k}{10}$ (Your answer should contain no summations but may still contain binomial coefficients/combinations).

Solution

Problem 4

Rice University and Stanford University write questions and corresponding solutions for a high school math tournament. The Rice group writes $10$ questions every hour but make a mistake in calculating their solutions $10\%$ of the time. The Stanford group writes $20$ problems every hour and makes solution mistakes $20\%$ of the time. Each school works for $10$ hours and then sends all problems to Smartie to be checked. However, Smartie isn’t really so smart, and only $75\%$ of the problems she thinks are wrong are actually incorrect. Smartie thinks $20\%$ of questions from Rice have incorrect solutions, and that $10\%$ 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?

Solution

Problem 5

Evaluate: $\sum_{k=1}^{\infty}\frac{1}{k\sqrt{k+2}+(k+2)\sqrt{k}}$

Solution

Problem 6

Ten teams of five runners each compete in a cross-country race. A runner finishing in $n\text{th}$ place contributes $n$ 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?

Solution

Problem 7

A lattice point in the plane is a point whose coordinates are both integers. Given a set of $100$ distinct lattice points in the plane, find the smallest number of line segments $\overline{AB}$ for which $A$ and $B$ are distinct lattice points in this set and the midpoint of $\overline{AB}$ is also a lattice point (not necessarily in the set).

Solution

Problem 8

The following computation arose in the research of mathematician P.D.: Let $k_i(j)=\frac{n+1}{2n+1}\frac{\binom{n}{i}\binom{n}{j}}{\binom{2n}{i+j}}$ for $0\le i, j\le n$ . $\sum_{j=0}^{n}k_i(j).$

Solution

Problem 9

How many positive integers appear in the list $\lfloor\frac{2006}{1}\rfloor, \lfloor\frac{2006}{2}\rfloor, \cdots, \lfloor\frac{2006}{2006}\rfloor$ where $\lfloor x\rfloor$ represents the greatest integer that does not exceed $x$ ?