Olympiad problems

2019 Teodor Topan, 2

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]

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2013 Bogdan Stan, 4

Let be a sequence $ \left( x_n \right)_{n\ge 1} $ having the property that $$ \lim_{n\to\infty } \left( 14(n+2)x_{n+2} -15(n+1)x_{n+1} +nx_n \right) =13. $$ Show that $ \left( x_n \right)_{n\ge 1} $ is convergent and calculate its limit. [i]Cosmin Nițu[/i]

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2007 Gheorghe Vranceanu, 4

Let be a sequence $ \left( a_n \right)_{n\geqslant 1} $ of real numbers defined recursively as $$ a_n=2007+1004n^2-a_{n-1}-a_{n-2}-\cdots -a_2-a_1. $$ Calculate: $$ \lim_{n\to\infty} \frac{1}{n}\int_1^{a_n} e^{1/\ln t} dt $$

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2005 Gheorghe Vranceanu, 4

$ \lim_{n\to\infty } \left( (1+1/n)^{-n}\sum_{i=0}^n\frac{1}{i!} \right)^{2n} $

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2006 Petru Moroșan-Trident, 2

Study the convergence of the sequence $$ \left( \sum_{k=2}^{n+1} \sqrt[k]{n+1} -\sum_{k=2}^{n} \sqrt[k]{n} \right)_{n\ge 2} , $$ and calculate its limit. [i]Dan Negulescu[/i]

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2007 Nicolae Coculescu, 2

Let be two sequences $ \left( a_n \right)_{n\ge 0} , \left( b_n \right)_{n\ge 0} $ satisfying the following system: $$ \left\{ \begin{matrix} a_0>0,& \quad a_{n+1} =a_ne^{-a_n} , &\quad\forall n\ge 0 \\ b_{0}\in (0,1) ,& \quad b_{n+1} =b_n\cos \sqrt{b_n} ,& \quad\forall n\ge 0 \end{matrix} \right. $$ Calculate $ \lim_{n\to\infty} \frac{a_n}{b_n} . $ [i]Florian Dumitrel[/i]

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