IMC 1996

Let $A=(a_{ij})\in M_{(n+1)\times (n+1)}(\mathbb{R})$ with $a_{ij}=a+|i-j|d$ , where $a$ and $d$ are fixed real numbers. Calculate $\det(A)$ .

Evaluate the definite integral $$\int_{-\pi}^{\pi}\frac{\sin nx}{(1+2^{x})\sin x} dx,$$ where $n$ is a natural number.

The linear operator $A$ on a finite-dimensional vector space $V$ is called an involution if $A^{2}=I$ , where $I$ is the identity operator. Let $\dim V=n$ . i) Prove that for every involution $A$ on $V$ , there exists a basis of $V$ consisting of eigenvectors of $A$ . ii) Find the maximal number of distinct pairwise commuting involutions on $V$ .

Let $a_{1}=1$ , $a_{n}=\frac{1}{n} \sum_{k=1}^{n-1}a_{k}a_{n-k}$ for $n\geq 2$ . Show that i) $\limsup_{n\to \infty} |a_{n}|^{\frac{1}{n}}<2^{-\frac{1}{2}}$ ; ii) $\limsup_{n\to \infty} |a_{n}|^{\frac{1}{n}}\geq \frac{2}{3}$

i) Let $a,b$ be real numbers such that $b\leq 0$ and $1+ax+bx^{2} \geq 0$ for every $x\in [0,1]$ . Prove that $$\lim_{n\to \infty} n \int_{0}^{1}(1+ax+bx^{2})^{n}dx= \begin{cases} -\frac{1}{a} &\text{if} a<0, \infty & \text{if} a \geq 0. \end{cases}$$ ii) Let $f:[0,1]\rightarrow[0,\infty)$ be a function with a continuous second derivative and let $f''(x)\leq0$ for every $x\in [0,1]$ . Suppose that $L=\lim_{n\to \infty} n \int_{0}^{1}(f(x))^{n}dx$ exists and $0<L<\infty$ . Prove that $f'$ has a constant sign and $\min_{x\in [0,1]}|f'(x)|=L^{-1}$ .

Upper content of a subset $E$ of the plane $\mathbb{R}^{2}$ is defined as $$\mathcal{C}(E)=\inf\{\sum_{i=1}^{n} \text{diam}(E_{i})\}$$ where $\inf$ is taken over all finite families of sets $E_{1},\dots,E_{n}$ $n\in \mathbb{N}$ , in $\mathbb{R}^{2}$ such that $E\subset \bigcup_{i=1}^{n}E_{i}$ . Lower content of $E$ is defined as $$\mathcal{K}(E)=\sup\{\text{length}(L) | L \text{ is a closed line segment onto which $E$ can be contracted}\}$$ . Prove that i) $\mathcal{C}(L)=\text{length}(L)$ if $L$ is a closed line segment; ii) $\mathcal{C}(E) \geq \mathcal{K}(E)$ ; iii) the equality in ii) is not always true even if $E$ is compact.

Prove that if $f:[0,1]\rightarrow[0,1]$ is a continuous function, then the sequence of iterates $x_{n+1}=f(x_{n})$ converges if and only if $$\lim_{n\to \infty}(x_{n+1}-x_{n})=0$$

Let $\theta$ be a positive real number. Show that if $k\in \mathbb{N}$ and both $\cosh k \theta$ and $\cosh(k+1) \theta$ are rational, then so is $\cosh \theta$ .

Let $G$ be the subgroup of $GL_{2}(\mathbb{R})$ generated by $A$ and $B$ , where $$A=\begin{pmatrix} 2 &0 0&1 \end{pmatrix}, B=\begin{pmatrix} 1 &1 0&1 \end{pmatrix}$$ . Let $H$ consist of the matrices $\begin{pmatrix} a_{11} &a_{12} a_{21}& a_{22} \end{pmatrix}$ in $G$ for which $a_{11}=a_{22}=1$ . a) Show that $H$ is an abelian subgroup of $G$ . b) Show that $H$ is not finitely generated.

Let $B$ be a bounded closed convex symmetric (with respect to the origin) set in $\mathbb{R}^{2}$ with boundary $\Gamma$ . Let $B$ have the property that the ellipse of maximal area contained in $B$ is the disc $D$ of radius $1$ centered at the origin with boundary $C$ . Prove that $A \cap \Gamma \ne \emptyset$ for any arc $A$ of $C$ of length $l(A)\geq \frac{\pi}{2}$ .