IMC 2010

Let $0 < a < b$. Prove that

Compute the sum of the series

Define the sequence $x_1, x_2, \dots$ inductively by $x_1 = \sqrt{5}$ and $x_{n+1} = x_n^2 - 2$ for each $n \ge 1$. Compute

Let $a$, $b$ be two integers and suppose that $n$ is a positive integer for which the set

is finite. Prove that $n = 1$.

Suppose that $a, b, c$ are real numbers in the interval $[-1,1]$ such that

Prove that

for all positive integers $n$.

(a) A sequence $x_1, x_2, \dots$ of real numbers satisfies

Does it follow that this sequence converges for all initial values $x_1$?

(b) A sequence $y_1, y_2, \dots$ of real numbers satisfies

Does it follow that this sequence converges for all initial values $y_1$?

Let $a_0, a_1, \dots, a_n$ be positive real numbers such that $a_{k+1} - a_k \ge 1$ for all $k = 0, 1, \dots, n-1$. Prove that

Denote by $S_n$ the group of permutations of the sequence $(1, 2, \dots, n)$. Suppose that $G$ is a subgroup of $S_n$, such that for every $\pi \in G \setminus \{e\}$ there exists a unique $k \in \{1, 2, \dots, n\}$ for which $\pi(k) = k$. (Here $e$ is the unit element in the group $S_n$.) Show that this $k$ is the same for all $\pi \in G \setminus \{e\}$.

Let $A$ be a symmetric $m \times m$ matrix over the two-element field all of whose diagonal entries are zero. Prove that for every positive integer $n$ each column of the matrix $A^n$ has a zero entry.

Suppose that for a function $f : \mathbb{R} \to \mathbb{R}$ and real numbers $a < b$ one has $f(x) = 0$ for all $x \in (a,b)$. Prove that $f(x) = 0$ for all $x \in \mathbb{R}$ if

for every prime number $p$ and every real number $y$.