Olympiad problems

2019 Teodor Topan, 2

Tags: squeeze theorem , seqeunce , newton s binom , cesaro-stolz , arithmetic sequence , real analysis , ratio

Let $ \left( a_n \right)_{n\ge 1} $ be an arithmetic progression with $ a_1=1 $ and natural ratio. [b]a)[/b] Prove that $$ a_n^{1/a_k} <1+\sqrt{\frac{2\left( a_n-1 \right)}{a_k\left( a_k -1 \right)}} , $$ for any natural numbers $ 2\le k\le n. $ [b]b)[/b] Calculate $ \lim_{n\to\infty } \frac{1}{a_n}\sum_{k=1}^n a_n^{1/a_k} . $ [i]Nicolae Bourbăcuț[/i]

2011 Laurențiu Duican, 2

Tags: squeeze theorem , definite integral , limit , integration , calculus , real analysis

$ \lim_{n\to\infty } \int_{\pi }^{2\pi } \frac{|\sin (nx) +\cos (nx)|}{ x} dx ? $ [i]Gabriela Boeriu[/i]

2008 Alexandru Myller, 1

Tags: integration , squeeze theorem , calculus

$ \lim_{n\to\infty} n2^n\int_1^n \frac{dx}{\left( 1+x^2\right)^n} $ [i][i]Bogdan Enescu[/i][/i]

2008 Alexandru Myller, 3

Tags: limit , squeeze theorem , sequence

Let be a $ \beta >1. $ Calculate $ \lim_{n\to\infty} \frac{k(n)}{n} ,$ where $ k(n) $ is the smallest natural number that satisfies the inequality $ (1+n)^k\ge n^k\beta . $ [i]Neculai Hârţan[/i]