CIIM 2013

Given two natural numbers $m$ and $n$ , denote by $\overline{m.n}$ the number obtained by writing $m$ followed by $n$ after the decimal dot.

a) Prove that there are infinitely many natural numbers $k$ such that for any of them the equation $\overline{m.n} \times \overline{n.m} = k$ has no solution.

b) Prove that there are infinitely many natural numbers $k$ such that for any of them the equation $\overline{m.n} \times \overline{n.m} = k$ has a solution.

Consider a polinomial $p \in \mathbb{R}[x]$ of degree $n$ and with no real roots. Prove that $$\int_{-\infty}^{\infty}\frac{(p'(x))^2}{(p(x))^2+(p'(x))^2}dx$$ converges, and is less or equal than $n^{3/2}\pi.$

Given a set of boys and girls, we call a pair $(A,B)$ amicable if $A$ and $B$ are friends. The friendship relation is symmetric. A set of people is affectionate if it satisfy the following conditions:

i) The set has the same number of boys and girls.

ii) For every four different people $A,B,C,D$ if the pairs $(A,B),(B,C),(C,D)$ and $(D,A)$ are all amicable, then at least one of the pairs $(A,C)$ and $(B,D)$ is also amicable. iii) At least $\frac{1}{2013}$ of all boy-girl pairs are amicable.

Let $m$ be a positive integer. Prove that there exists an integer $N(m)$ such that if a affectionate set has al least $N(m)$ people, then there exists $m$ boys that are pairwise friends or $m$ girls that are pairwise friends.

Let $a_1,b_1,c_1,a_2,b_2,c_2$ be positive real number and $F,G:(0,\infty)\to(0,\infty)$ be to differentiable and positive functions that satisfy the identities: $$\frac{x}{F} = 1 + a_1x+ b_1y + c_1G$$ $$\frac{y}{G} = 1 + a_2x+ b_2y + c_2F.$$ Prove that if $0 < x_1 \leq x_2$ and $0 < y_2 \leq y_1$ , then $F(x_1,x_2) \leq F(x_2,y_2)$ and $G(x_1,y_1) \geq G(x_2,y_2).$

Let $A,B$ be $n\times n$ matrices with complex entries. Show that there exists a matrix $T$ and an invertible matrix $S$ such that $$B=S(A+T)S^{-1}\ -T \iff \operatorname{tr}(A) = \operatorname{tr}(B)$$

Let $(X,d)$ be a metric space with $d:X\times X \to \mathbb{R}_{\geq 0}$ . Suppose that $X$ is connected and compact. Prove that there exists an $\alpha \in \mathbb{R}_{\geq 0}$ with the following property: for any integer $n > 0$ and any $x_1,\dots,x_n \in X$ , there exists $x\in X$ such that the average of the distances from $x_1,\dots,x_n$ to $x$ is $\alpha$ i.e. $$\frac{d(x,x_1)+d(x,x_2)+\cdots+d(x,x_n)}{n} = \alpha.$$