International Mathematics Competition
for University Students
2015

IMC2015: Problems on Day 2

6. Prove that $$\sum\limits_{n = 1}^{\infty}\frac{1}{\sqrt{n}\left(n+1\right)} < 2.$$

Proposed by Ivan Krijan, University of Zagreb

  Solution  

7. Compute $$ \lim_{A\to+\infty}\frac1A\int_1^A A^{\frac1x}\dx\,. $$

Proposed by Jan Šustek, University of Ostrava

  Hint    Solution  

8. 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

  Solution  

9. 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

  Solution  

10. Let $n$ be a positive integer, and let $p(x)$ be a polynomial of degree $n$ with integer coefficients. Prove that $$ \max_{0\le x\le1} \big|p(x)\big| > \frac1{e^n}. $$

Proposed by Géza Kós, Eötvös University, Budapest

  Hint    Solution