IMC 2003

(a) Let $a_1, a_2, \dots$ be a sequence of real numbers such that $a_1 = 1$ and $a_{n+1} > \frac{3}{2}a_n$ for all $n$. Prove that the sequence

has a finite limit or tends to infinity.

(b) Prove that for all $\alpha > 1$ there exists a sequence $a_1, a_2, \dots$ with the same properties such that

Let $a_1, a_2, \dots, a_{51}$ be non-zero elements of a field. We simultaneously replace each element with the sum of the $50$ remaining ones. In this way we get a sequence $b_1, \dots, b_{51}$. If this new sequence is a permutation of the original one, what can be the characteristic of the field? (The characteristic of a field is $p$, if $p$ is the smallest positive integer such that $\underbrace{x + x + \dots + x}_{p} = 0$ for any element $x$ of the field. If there exists no such $p$, the characteristic is $0$.)

Let $A$ be an $n \times n$ real matrix such that $3A^3 = A^2 + A + I$ ($I$ is the identity matrix). Show that the sequence $A^k$ converges to an idempotent matrix. (A matrix $B$ is called idempotent if $B^2 = B$.)

Determine the set of all pairs $(a,b)$ of positive integers for which the set of positive integers can be decomposed into two sets $A$ and $B$ such that $a \cdot A = b \cdot B$.

Let $g : [0,1] \to \mathbb{R}$ be a continuous function and let $f_n : [0,1] \to \mathbb{R}$ be a sequence of functions defined by $f_0(x) = g(x)$ and

Determine $\lim\limits_{n \to \infty} f_n(x)$ for every $x \in (0,1]$.

Let $f(z) = a_n z^n + a_{n-1}z^{n-1} + \dots + a_1 z + a_0$ be a polynomial with real coefficients. Prove that if all roots of $f$ lie in the left half-plane $\{z \in \mathbb{C} : \operatorname{Re} z < 0\}$ then

holds for every $k = 0, 1, \dots, n-3$.

Let $A$ and $B$ be $n \times n$ real matrices such that $AB + A + B = 0$. Prove that $AB = BA$.

Evaluate the limit

Let $A$ be a closed subset of $\mathbb{R}^n$ and let $B$ be the set of all those points $b \in \mathbb{R}^n$ for which there exists exactly one point $a_0 \in A$ such that

Prove that $B$ is dense in $\mathbb{R}^n$; that is, the closure of $B$ is $\mathbb{R}^n$.

Find all positive integers $n$ for which there exists a family $\mathcal{F}$ of three-element subsets of $S = \{1, 2, \dots, n\}$ satisfying the following two conditions:

(i) for any two different elements $a, b \in S$, there exists exactly one $A \in \mathcal{F}$ containing both $a$, $b$;

(ii) if $a, b, c, x, y, z$ are elements of $S$ such that if $\{a,b,x\}, \{a,c,y\}, \{b,c,z\} \in \mathcal{F}$, then $\{x,y,z\} \in \mathcal{F}$.

(a) Show that for each function $f : \mathbb{Q} \times \mathbb{Q} \to \mathbb{R}$ there exists a function $g : \mathbb{Q} \to \mathbb{R}$ such that $f(x,y) \le g(x) + g(y)$ for all $x, y \in \mathbb{Q}$.

(b) Find a function $f : \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ for which there is no function $g : \mathbb{R} \to \mathbb{R}$ such that $f(x,y) \le g(x) + g(y)$ for all $x, y \in \mathbb{R}$.

Let $(a_n)_{n \in \mathbb{N}}$ be the sequence defined by

Find the limit

if it exists.