## IMC 2015 for University Student: Day 2

Problem 6. Prove that

$\displaystyle \sum_{n=1}^\infty \frac{1}{\sqrt{n}(n+1)} < 2.$

Problem 7. Compute

$\displaystyle \lim_{A \rightarrow \infty} \frac{1}{A} \int_1^A A^{\frac{1}{x}} dx.$

Problem 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 weight of all words is $3^{75}$.

Problem 9. An $n \times n$ complex matrix $A$ is called t-normal if $A^tA = AA^t$ 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.

Problem 10. Let $n$ be a positive integer and let $p(x)$ be a polynomial of degree $n$ with integer coefficients. Prove that

$\text{max}_{0 \le x \le 1} |p(x)| > \frac{1}{e^n}.$