VTRMC 2010

Let $d$ be a positive integer and let $A$ be a $d\times d$ matrix with integer entries. Suppose $I+A+A_2+\ldots+A_{100}=0$ (where $I$ denotes the identity $d\times d$ matrix, and $0$ denotes the zero matrix, which has all entries $0$ ). Determine the positive integers $n\le100$ for which $A_n+A_{n+1}+\ldots+A_{100}$ has determinant $\pm1$ .

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 $f_{100}(75)\pmod{17}$ . Justify your answer.

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 $(a-b)^2(a+b)=bc^2$ .

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?

Define a sequence by $a_1=1,a_2=\frac12$ , and $a_{n+2}=a_{n+1}-\frac{a_na_{n+1}}2$ for $n$ a positive integer. Find $\lim_{n\to\infty}na_n$ .

Let $\sum_{n=1}^\infty a_n$ be a convergent series of positive terms (so $a_i>0$ for all $i$ ) and set $b_n=\frac1{na_n^2}$ for $n\ge1$ . Prove that $\sum_{n=1}^\infty\frac n{b_1+b_2+\ldots+b_n}$ is convergent.