IMC 2014

Determine all pairs $(a,b)$ of real numbers for which there exists a unique symmetric $2 \times 2$ matrix $M$ with real entries satisfying $\operatorname{trace}(M) = a$ and $\det(M) = b$.

Consider the following sequence

Find all pairs $(\alpha, \beta)$ of positive real numbers such that $\lim\limits_{n \to \infty} \dfrac{\sum\limits_{k=1}^{n} a_k}{n^{\alpha}} = \beta$.

Let $n$ be a positive integer. Show that there are positive real numbers $a_0, a_1, \dots, a_n$ such that for each choice of signs the polynomial

has $n$ distinct real roots.

Let $n > 6$ be a perfect number, and let $n = p_1^{e_1} \cdots p_k^{e_k}$ be its prime factorisation with $1 < p_1 < \dots < p_k$. Prove that $e_1$ is an even number.

A number $n$ is perfect if $s(n) = 2n$, where $s(n)$ is the sum of the divisors of $n$.

For a positive integer $x$, denote its $n^{\text{th}}$ decimal digit by $d_n(x)$, i.e. $d_n(x) \in \{0, 1, \dots, 9\}$ and $x = \sum\limits_{n=1}^{\infty} d_n(x)10^{n-1}$. Suppose that for some sequence $(a_n)_{n=1}^{\infty}$, there are only finitely many zeros in the sequence $\big(d_n(a_n)\big)_{n=1}^{\infty}$. Prove that there are infinitely many positive integers that do not occur in the sequence $(a_n)_{n=1}^{\infty}$.

Let $A = (a_{ij})_{i,j=1}^{n}$ be a symmetric $n \times n$ matrix with real entries, and let $\lambda_1, \lambda_2, \dots, \lambda_n$ denote its eigenvalues. Show that

and determine all matrices for which equality holds.

Let $f(x) = \dfrac{\sin x}{x}$, for $x > 0$, and let $n$ be a positive integer. Prove that $\left| f^{(n)}(x) \right| < \dfrac{1}{n+1}$, where $f^{(n)}$ denotes the $n^{\text{th}}$ derivative of $f$.

We say that a subset of $\mathbb{R}^n$ is $k$-almost contained by a hyperplane if there are less than $k$ points in that set which do not belong to the hyperplane. We call a finite set of points $k$-generic if there is no hyperplane that $k$-almost contains the set. For each pair of positive integers $k$ and $n$, find the minimal number $d(k,n)$ such that every finite $k$-generic set in $\mathbb{R}^n$ contains a $k$-generic subset with at most $d(k,n)$ elements.

For every positive integer $n$, denote by $D_n$ the number of permutations $(x_1, \dots, x_n)$ of $(1, 2, \dots, n)$ such that $x_j \ne j$ for every $1 \le j \le n$. For $1 \le k \le \frac{n}{2}$, denote by $\Delta(n,k)$ the number of permutations $(x_1, \dots, x_n)$ of $(1, 2, \dots, n)$ such that $x_i = k + i$ for every $1 \le i \le k$ and $x_j \ne j$ for every $1 \le j \le n$. Prove that