For any integer $n \ge 2$ and two $n \times n$ matrices with real entries $A$, $B$ that satisfy the equation $$A^{-1} + B^{-1} = (A + B)^{-1}$$ prove that $\det(A) = \det(B)$.

Does the same conclusion follow for matrices with complex entries?

(Proposed by Zbigniew Skoczylas, Wrocł aw University of Technology)

For a positive integer $n$, let $f(n)$ be the number obtained by writing $n$ in binary and replacing every 0 with 1 and vice versa. For example, $n = 23$ is 10111 in binary, so $f(n)$ is 1000 in binary, therefore $f(23) = 8$. Prove that $$\sum_{k=1}^{n} f(k) \le \frac{n^2}{4}.$$ When does equality hold?

(Proposed by Stephan Wagner, Stellenbosch University)

Let $F(0) = 0$, $F(1) = \frac{3}{2}$, and $F(n) = \frac{5}{2} F(n-1) - F(n-2)$ for $n \ge 2$.

Determine whether or not $\displaystyle\sum_{n=0}^{\infty} \frac{1}{F(2^n)}$ is a rational number.

(Proposed by Gerhard Woeginger, Eindhoven University of Technology)

Determine whether or not there exist 15 integers $m_1, \dots, m_{15}$ such that $$\tag{1} \sum_{k=1}^{15} m_k \cdot \arctan(k) = \arctan(16).$$ (Proposed by Gerhard Woeginger, Eindhoven University of Technology)

Let $n \ge 2$, let $A_1, A_2, \dots, A_{n+1}$ be $n + 1$ points in the $n$-dimensional Euclidean space, not lying on the same hyperplane, and let $B$ be a point strictly inside the convex hull of $A_1, A_2, \dots, A_{n+1}$. Prove that $\angle A_i B A_j > 90^\circ$ holds for at least $n$ pairs $(i, j)$ with $1 \le i < j \le n + 1$.

(Proposed by Géza Kós, Eötvös University, Budapest)

Prove that $$\sum_{n=1}^{\infty} \frac{1}{\sqrt{n(n+1)}} < 2.$$ (Proposed by Ivan Krijan, University of Zagreb)

Compute $$\lim_{A \to +\infty} \frac{1}{A} \int_1^A A^{\frac{1}{x}}\,dx.$$ (Proposed by Jan Šustek, University of Ostrava)

Consider all $26^{26}$ words of length 26 in the Latin alphabet. Define the weight of a word as $1/(k+1)$, where $k$ is the number of letters not used in this word. Prove that the sum of the weights of all words is $3^{75}$.

(Proposed by Fedor Petrov, St. Petersburg State University)

An $n \times n$ complex matrix $A$ is called t-normal if $A A^t = A^t A$ where $A^t$ is the transpose of $A$. For each $n$, determine the maximum dimension of a linear space of complex $n \times n$ matrices consisting of t-normal matrices.

(Proposed by Shachar Carmeli, Weizmann Institute of Science)

Let $n$ be a positive integer, and let $p(x)$ be a polynomial of degree $n$ with integer coefficients. Prove that $$\max_{0 \le x \le 1} \bigl| p(x) \bigr| > \frac{1}{e^n}.$$ (Proposed by Géza Kós, Eötvös University, Budapest)