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}.

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