Olympiad problems

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Found problems: 3

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2019 Centers of Excellency of Suceava, 2

Let be two real numbers $ b>a>0, $ and a sequence $ \left( x_n \right)_{n\ge 1} $ with $ x_2>x_1>0 $ and such that $$ ax_{n+2}+bx_n\ge (a+b)x_{n+1} , $$ for any natural numbers $ n. $ Prove that $ \lim_{n\to\infty } x_n=\infty . $ [i]Dan Popescu[/i]

2012 Bogdan Stan, 2

Tags: sequence , real analyssi

Let be a bounded sequence of positive real numbers $ \left( x_n \right)_{n\ge 1} $ satisfying the recurrence: $$ x_{n+3} =\sqrt[3]{3x_n-2} . $$ Prove that $ \left( x_n \right)_{n\ge 1} $ is convergent. [i]Cristinel Mortici[/i]

2012 Bogdan Stan, 3

Tags: function , find all functions , real analyssi

Find all functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that verify the equality $$ \int_a^b f(x)dx=f(b)-f(a), $$ for any real numbers $ a,b. $ [i]Cosmin Nitu[/i]

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