Virginia Tech Regional Math Contest (2016)

Problem 1

Evaluate $\displaystyle \int_1^2 \frac{\ln x}{2 - 2x + x^2} dx$.

Problem 2

Determine the real numbers $k$ such that $\displaystyle \sum_{n=1}^\infty \left( \frac{(2n)!}{4^n n! n!} \right)^k$ is convergent.

Problem 3

Let $n$ be a positive integer and let $M_n (\mathbb{Z}_2)$ denote the $n$ by $n$ matrices with entries from the integers mod 2. If $n \ge 2$, prove that the number of matrices $A$ in $M_n (\mathbb{Z}_2)$ satisfying $A^2 = 0$ (the matrix with all entries zero) is an even positive integer.

Problem 4

For a positive integer $a$, let $P(a)$ denote the largest prime divisor of $a^2 + 1$. Prove that there exist infinitely many triples $(a, b, c)$ of distinct positive integers such that $P(a) = P(b) = P(c)$.

Problem 5

Suppose that $m, n, r$ are positive integers such that $$1 + m + n \sqrt{3} = (2 + \sqrt{3})^{2r-1}$$ Prove that $m$ is a perfect square.

Problem 6

Let $A, B, P, Q, X, Y$ be square matrices of the same size. Suppose that

$$\begin{align*} A + B + AB & = XY & \hspace{1cm} AX & = XQ \newline P + Q + PQ & = YX & \hspace{1cm} PY & = YB. \end{align*}$$

Prove that $AB = BA$.

Problem 7

Let $q$ be a real number with $|q| \ne 1$ and let $k$ be a positive integer. Define a Laurent polynomial $f_k (X)$ in the variable $X$, depending on $q$ and $k$, by

$$f_k (X) = \prod_{i=0}^{k-1} (1 - q^i X)(1 - q^{i+1} X^{-1}).$$

Show that the constant term of $f_k (X)$, i.e., the coefficient of $X^0$ in $f_k (X)$, is equal to

$$\frac{(1 - q^{k+1})(1 - q^{k+2}) \cdots (1 - q^{2k})}{(1 - q)(1 - q^2) \cdots (1 - q^k)}.$$