Virginia Tech Regional Math Contest (2010)

Problem 1

Let $d$ be a positive integer and let $A$ be a $d \times d$ matrix with integer entries. Suppose $I + A + A^2 + \dots + A^{100} = 0$ (where $I$ denotes the identity $d \times d$ matrix, so $I$ has 1’s on the main diagonal, and $0$ denotes the zero matrix, which has all entries 0). Determine the positive integers $n \leq 100$ for which $A^n + A^{n+1} + \dots + A^{100}$ has determinant $\pm 1$.

Problem 2

For $n$ a positive integer, define $f_1 (n) = n$ and then for $i$ a positive integer, define $f_{i+1} (n) = f_i (n)^{f_i (n)}$. Determine $f_{100} (75) \bmod 17$ (i.e. determine the remainder after dividing $f_{100} (75)$ by 17, an integer between 0 and 16). Justify your answer.

Problem 3

Prove that $\cos(\pi/7)$ is a root of the equation $8x^3 - 4x^2 - 4x + 1 = 0$, and find the other two roots.

Problem 4

Let $\triangle ABC$ be a triangle with sides $a, b, c$ and corresponding angles $A, B, C$ (so $a = BC$ and $A = \angle BAC$ etc.). Suppose that $4A + 3C = 540^\circ$. Prove that $(a - b)^2 (a + b) = b c^2$.

Problem 5

Let $A, B$ be two circles in the plane with $B$ inside $A$. Assume that $A$ has radius 3, $B$ has radius 1, $P$ is a point on $A$, $Q$ is a point on $B$, and $A$ and $B$ touch so that $P$ and $Q$ are the same point. Suppose that $A$ is kept fixed and $B$ is rolled once round the inside of $A$ so that $Q$ traces out a curve starting and finishing at $P$. What is the area enclosed by this curve?

Problem 6

Define a sequence by $a_1 = 1$, $a_2 = 1/2$, and $a_{n+2} = a_{n+1} - \dfrac{a_n a_{n+1}}{2}$ for $n$ a positive integer. Find $\displaystyle \lim_{n \to \infty} n a_n$.

Problem 7

Let $\displaystyle \sum_{n=1}^\infty a_n$ be a convergent series of positive terms (so $a_i > 0$ for all $i$) and set $b_n = \dfrac{1}{n a_n^2}$ for $n \geq 1$. Prove that $\displaystyle \sum_{n=1}^\infty \frac{n}{b_1 + b_2 + \dots + b_n}$ is convergent.