IMC 2015

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, Wroclaw 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 $$\sum_{k=1}^n f(k) \leq \frac{n^2}{4}.$$ When does equality hold?

(Proposed by Stephan Wagner, Stellenbosch University)

Let $F(0)=0$ , $F(1)=\frac32$ , and $F(n)=\frac{5}{2}F(n-1)-F(n-2)$ for $n\ge2$ .

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

(Proposed by Gerhard Woeginger, Eindhoven University of Technology)

Compute $$\lim_{A\to+\infty}\frac1A\int_1^A A^{\frac1x} dx .$$ Proposed by Jan Šustek, University of Ostrava

Consider all $26^{26}$ words of length 26 in the Latin alphabet. Define the $\emph{weight}$ 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 \emph{t-normal} if $AA^t = A^t A$ 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