VTRMC 2006

Find, with proof, all positive integers $n$ such that neither $n$ nor $n^2$ contain a $1$ when written in base $3$ .

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 $S(10)$ .

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

(a) $p$ and $q$ are continuous functions on the open interval $(0,\pi)$ . (b) $f$ is an infinitely differentiable nonzero function on the whole real line $(-\infty,\infty)$ such that $f(0)=f'(0)=f''(0)$ . (c) $y=\sin t$ and $y=f(t)$ are solutions of the differential equation $y''+p(t)y'+q(t)y=0$ on $(0,\pi)$ .

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

Let $\{a_n\}$ be a monotonically decreasing sequence of positive real numbers with limit $0$ . Let $\{b_n\}$ 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 $a_{3m+1}$ , $a_{3m+2}$ , $a_{3m+3}$ . Prove or give a counterexample to the following statement: the series $\sum_{n=1}^\infty(-1)^nb_n$ is convergent.

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 $AB+AC=3BC$ .

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 $M$ .