SEEMOUS 2012

Let $A=(a_{ij})$ be the $n\times n$ matrix, where $a_{ij}$ is the remainder of the division of $i^j+j^i$ by $3$ for $i,j=1,2,\ldots,n$ . Find the greatest $n$ for which $\det A\ne0$ .

Let $a_n>0$ , $n\ge1$ . Consider the right triangles $\triangle A_0A_1A_2$ , $\triangle A_0A_2A_3,\ldots$ , $\triangle A_0A_{n-1}A_n,\ldots,$ as in the figure. (More precisely, for every $n\ge2$ the hypotenuse $A_0A_n$ of $\triangle A_0A_{n-1}A_n$ is a leg of $\triangle A_0A_nA_{n+1}$ with right angle $\angle A_0A_nA_{n+1}$ , and the vertices $A_{n-1}$ and $A_{n+1}$ lie on the opposite sides of the straight line $A_0A_n$ ; also, $|A_{n-1}A_n|=a_n$ for every $n\ge1$ .) Is it possible for the set of points $\{A_n\mid n\ge0\}$ to be unbounded but the series $\sum_{n=2}^\infty m\angle A_{n-1}A_0A_n$ to be convergent? Note. A subset $B$ of the plane is bounded if and only if there is a disk $D$ such that $B\subseteq D$ .

a) Prove that if $k$ is an even positive integer and $A$ is a real symmetric $n\times n$ matrix such that $\operatorname{tr}(A^k)^{k+1}=\operatorname{tr}(A^{k+1})^k$ , then $$A^n=\operatorname{tr}(A)A^{n-1}.$$ b) Does the assertion from a) also hold for odd positive integers $k$ ?

a) Compute $$\lim_{n\to\infty}n\int^1_0\left(\frac{1-x}{1+x}\right)^ndx.$$ b) Let $k\ge1$ be an integer. Compute $$\lim_{n\to\infty}n^{k+1}\int^1_0\left(\frac{1-x}{1+x}\right)^nx^kdx.$$