OIMU 2005

Let $P(x,y)=(x^2y^3,x^3y^5)$ , $P^1=P$ and $P^{n+1}=P\circ P^n$ . Also, let $p_n(x)$ be the first coordinate of $P^n(x,x)$ , and $f(n)$ be the degree of $p_n(x)$ . Find $$\lim_{n\to\infty}f(n)^{1/n}$$

Let $A,B,C$ be real square matrices of order $n$ such that $A^3=-I$ , $BA^2+BA=C^6+C+I$ and $C$ is symmetric. Is it possible that $n=2005$ ?

Consider the sequence defined recursively by $(x_1,y_1)=(0,0)$ , $(x_{n+1},y_{n+1})=\left(\left(1-\frac{2}{n}\right)x_n-\frac{1}{n}y_n+\frac{4}{n},\left(1-\frac{1}{n}\right)y_n-\frac{1}{n}x_n+\frac{3}{n}\right)$ .

Find $\lim_{n\to \infty}(x_n,y_n)$ .

A variable tangent $t$ to the circle $C_1$ , of radius $r_1$ , intersects the circle $C_2$ , of radius $r_2$ in $A$ and $B$ . The tangents to $C_2$ through $A$ and $B$ intersect in $P$ . Find, as a function of $r_1$ and $r_2$ , the distance between the centers of $C_1$ and $C_2$ such that the locus of $P$ when $t$ varies is contained in an equilateral hyperbola.Note: A hyperbola is said to be equilateral if its asymptotes are perpendicular.

Arnaldo and Bernaldo play a game where they alternate saying natural numbers, and the winner is the one who says $0$ . In each turn except the first the possible moves are determined from the previous number $n$ in the following way: write $$n =\sum_{m\in O_n}2^m;$$ the valid numbers are the elements $m$ of $O_n$ . That way, for example, after Arnaldo says $42= 2^5 + 2^3 + 2^1$ , Bernaldo must respond with $5$ , $3$ or $1$ .

We define the sets $A,B\subset \mathbb{N}$ in the following way. We have $n\in A$ iff Arnaldo, saying $n$ in his first turn, has a winning strategy; analogously, we have $n\in B$ iff Bernaldo has a winning strategy if Arnaldo says $n$ during his first turn. This way, $$A =\{0, 2, 8, 10,\cdots\}, B = \{1, 3, 4, 5, 6, 7, 9,\cdots\}$$ Define $f:\mathbb{N}\to \mathbb{N}$ by $f(n)=|A\cap \{0,1,\cdots,n-1\}|$ . For example, $f(8) = 2$ and $f(11)=4$ . Find $$\lim_{n\to\infty}\frac{f(n)\log(n)^{2005}}{n}$$

A smooth function $f:I\to \mathbb{R}$ is said to be totally convex if $(-1)^k f^{(k)}(t) > 0$ for all $t\in I$ and every integer $k>0$ (here $I$ is an open interval).

Prove that every totally convex function $f:(0,+\infty)\to \mathbb{R}$ is real analytic.Note: A function $f:I\to \mathbb{R}$ is said to be smooth if for every positive integer $k$ the derivative of order $k$ of $f$ is well defined and continuous over $\mathbb{R}$ . A smooth function $f:I\to \mathbb{R}$ is said to be real analytic if for every $t\in I$ there exists $\epsilon> 0$ such that for all real numbers $h$ with $|h|<\epsilon$ the Taylor series $$\sum_{k\geq 0}\frac{f^{(k)}(t)}{k!}h^k$$ converges and is equal to $f(t+h)$ .

Prove that for any integers $n,p$ , $0<n\leq p$ , all the roots of the polynomial below are real: $$P_{n,p}(x)=\sum_{j=0}^n {p\choose j}{p\choose {n-j}}x^j$$