IMC 1995

Let $X$ be a invertible matrix with columns $X_{1},X_{2}...,X_{n}$ . Let $Y$ be a matrix with columns $X_{2},X_{3},...,X_{n},0$ . Show that the matrices $A=YX^{-1}$ and $B=X^{-1}Y$ have rank $n-1$ and have only $0$ ´s for eigenvalues.

Let $f$ be a continuous function on $[0,1]$ such that for every $x\in [0,1]$ , we have $\int_{x}^{1}f(t)dt \geq\frac{1-x^{2}}{2}$ . Show that $\int_{0}^{1}f(t)^{2}dt \geq \frac{1}{3}$ .

Let $f$ be twice continuously differentiable on $(0,\infty)$ such that $\lim_{x \to 0^{+}}f'(x)=-\infty$ and $\lim_{x \to 0^{+}}f''(x)=\infty$ . Show that $$\lim_{x\to 0^{+}}\frac{f(x)}{f'(x)}=0.$$

Let $F:(1,\infty) \rightarrow \mathbb{R}$ be the function defined by $$F(x)=\int_{x}^{x^{2}} \frac{dt}{\ln(t)}.$$ Show that $F$ is injective and find the set of values of $F$ .

Let $A$ and $B$ be real $n\times n$ matrices. Assume there exist $n+1$ different real numbers $t_{1},t_{2},\dots,t_{n+1}$ such that the matrices $$C_{i}=A+t_{i}B, \, i=1,2,\dots,n+1$$ are nilpotent. Show that both $A$ and $B$ are nilpotent.

Let $p>1$ . Show that there exists a constant $K_{p} >0$ such that for every $x,y\in \mathbb{R}$ with $|x|^{p}+|y|^{p}=2$ , we have $$(x-y)^{2} \leq K_{p}(4-(x+y)^{2}).$$

Let $A$ be a $3\times 3$ real matrix such that the vectors $Au$ and $u$ are orthogonal for every column vector $u\in \mathbb{R}^{3}$ . Prove that: a) $A^{T}=-A$ . b) there exists a vector $v \in \mathbb{R}^{3}$ such that $Au=v\times u$ for every $u\in \mathbb{R}^{3}$ , where $v \times u$ denotes the vector product in $\mathbb{R}^{3}$ .

Let $(b_{n})_{n\in \mathbb{N}}$ be a sequence of positive real numbers such that $b_{0}=1$ , $b_{n}=2+\sqrt{b_{n-1}}-2\sqrt{1+\sqrt{b_{n-1}}}$ . Calculate $$\sum_{n=1}^{\infty}b_{n}2^{n}.$$

Let all roots of an $n$ -th degree polynomial $P(z)$ with complex coefficients lie on the unit circle in the complex plane. Prove that all roots of the polynomial $$2zP'(z)-nP(z)$$ lie on the same circle.

a) Prove that for every $\epsilon>0$ there is a positive integer $n$ and real numbers $\lambda_{1},\dots,\lambda_{n}$ such that $$\max_{x\in [-1,1]}|x-\sum_{k=1}^{n}\lambda_{k}x^{2k+1}|<\epsilon.$$ b) Prove that for every odd continuous function $f$ on $[-1,1]$ and for every $\epsilon>0$ there is a positive integer $n$ and real numbers $\mu_{1},\dots,\mu_{n}$ such that $$\max_{x\in [-1,1]}|f(x)-\sum_{k=1}^{n}\mu_{k}x^{2k+1}|<\epsilon.$$