IMC 2017

Determine all complex numbers $\lambda$ for which there exist a positive integer $n$ and a real $n \times n$ matrix $A$ such that $A^2 = A^T$ and $\lambda$ is an eigenvalue of $A$.

Let $f \colon \mathbb{R} \to (0, \infty)$ be a differentiable function, and suppose that there exists a constant $L > 0$ such that

for all $x, y$. Prove that

holds for all $x$.

For any positive integer $m$, denote by $P(m)$ the product of positive divisors of $m$ (e.g. $P(6) = 36$). For every positive integer $n$ define the sequence

Determine whether for every set $S \subseteq \{1, 2, \ldots, 2017\}$, there exists a positive integer $n$ such that the following condition is satisfied:

For every $k$ with $1 \le k \le 2017$, the number $a_k(n)$ is a perfect square if and only if $k \in S$.

There are $n$ people in a city, and each of them has exactly 1000 friends (friendship is always symmetric). Prove that it is possible to select a group $S$ of people such that at least $n/2017$ persons in $S$ have exactly two friends in $S$.

Let $k$ and $n$ be positive integers with $n \ge k^2 - 3k + 4$, and let

be a polynomial with complex coefficients such that

Prove that $f(z)$ and $z^n - 1$ have at most $n - k$ common roots.

Let $f \colon [0; +\infty) \to \mathbb{R}$ be a continuous function such that $\lim\limits_{x \to +\infty} f(x) = L$ exists (it may be finite or infinite). Prove that

Let $p(x)$ be a nonconstant polynomial with real coefficients. For every positive integer $n$, let

Prove that there are only finitely many numbers $n$ such that all roots of $q_n(x)$ are real.

Define the sequence $A_1, A_2, \ldots$ of matrices by the following recurrence:

where $I_m$ is the $m \times m$ identity matrix.

Prove that $A_n$ has $n + 1$ distinct integer eigenvalues $\lambda_0 < \lambda_1 < \ldots < \lambda_n$ with multiplicities $\binom{n}{0}, \binom{n}{1}, \ldots, \binom{n}{n}$, respectively.

Define the sequence $f_1, f_2, \ldots \colon [0, 1) \to \mathbb{R}$ of continuously differentiable functions by the following recurrence:

Show that $\lim\limits_{n \to \infty} f_n(x)$ exists for every $x \in [0, 1)$ and determine the limit function.

Let $K$ be an equilateral triangle in the plane. Prove that for every $p > 0$ there exists an $\varepsilon > 0$ with the following property: If $n$ is a positive integer, and $T_1, \ldots, T_n$ are non-overlapping triangles inside $K$ such that each of them is homothetic to $K$ with a negative ratio, and

then