Virginia Tech Regional Math Contest (2006)

Problem 1

Find, and give a proof of your answer, all positive integers $n$ such that neither $n$ nor $n^2$ contain a 1 when written in base 3.

Problem 2

Let $S(n)$ denote the number of sequences of length $n$ formed by the three letters A, B, C with the restriction that the C’s (if any) all occur in a single block immediately following the first B (if any). For example ABCCAA, AAABAA, and ABCCCC are counted in, but ACACCB and CAAAAA are not. Derive a simple formula for $S(n)$ and use it to calculate $S(10)$.

Problem 3

Recall that the Fibonacci numbers $F(n)$ are defined by $F(0) = 0$, $F(1) = 1$, and $F(n) = F(n - 1) + F(n - 2)$ for $n \ge 2$. Determine the last digit of $F(2006)$ (e.g., the last digit of 2006 is 6).

Problem 4

We want to find functions $p(t), q(t), f(t)$ such that

Is this possible? Either prove this is not possible, or show this is possible by providing an explicit example of such $f, p, q$.

Problem 5

Let ${a_n}$ be a monotonic decreasing sequence of positive real numbers with limit 0 (so $a_1 \ge a_2 \ge \dots \ge 0$). Let $\lbrace b_n \rbrace$ be a rearrangement of the sequence such that for every non-negative integer $m$, the terms $b_{3m+1}, b_{3m+2}, b_{3m+3}$ are a rearrangement of the terms $a_{3m+1}, a_{3m+2}, a_{3m+3}$ (thus, for example, the first 6 terms of the sequence $\lbrace b_n \rbrace$ could be $a_3, a_2, a_1, a_4, a_6, a_5$). Prove or give a counterexample to the following statement: the series $\displaystyle{\sum_{n=1}^{\infty} (-1)^n b_n}$ is convergent.

Problem 6

In the diagram below $BP$ bisects $\angle ABC$, $CP$ bisects $\angle BCA$, and $PQ$ is perpendicular to $BC$. If $BQ \cdot QC = 2PQ^2$, prove that $AB + AC = 3BC$.

Problem 7

Three spheres each of unit radius have centers $P, Q, R$ with the property that the center of each sphere lies on the surface of the other two spheres. Let $C$ denote the cylinder with cross-section $PQR$ (the triangular lamina with vertices $P, Q, R$) and axis perpendicular to $PQR$. Let $M$ denote the space which is common to the three spheres and the cylinder $C$, and suppose the mass density of $M$ at a given point is the distance of the point from $PQR$. Determine the mass of $M$.