Olympiad problems

2018 Brazil Undergrad MO, 10

How many ordered pairs of real numbers $ (a, b) $ satisfy equality $\lim_{x \to 0} \frac{\sin^2x}{e^{ax}-2bx-1}= \frac{1}{2}$?

2020 Brazil Undergrad MO, Problem 4

For each of the following, provide proof or a counterexample: a) Every $2\times2$ matrix with real entries can we written as the sum of the squares of two $2\times2$ matrices with real entries. b) Every $3\times3$ matrix with real entries can we written as the sum of the squares of two $3\times3$ matrices with real entries.

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2018 Brazil Undergrad MO, 22

What is the value of the improper integral $ \int_0 ^ {\pi} \log (\sin (x)) dx$?

calculus college contest integration

2018 IMC, 7

Let $(a_n)_{n=0}^{\infty}$ be a sequence of real numbers such that $a_0=0$ and $$a_{n+1}^3=a_n^2-8\quad \text{for} \quad n=0,1,2,…$$ Prove that the following series is convergent: $$\sum_{n=0}^{\infty}{|a_{n+1}-a_n|}.$$ [i]Proposed by Orif Ibrogimov, National University of Uzbekistan[/i]

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