This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 2

2010 Laurențiu Panaitopol, Tulcea, 1
Tags: limit , nested radical , weierstrass , real analysis , sequence
Show that if $ \left( s_n \right)_{n\ge 0} $ is a sequence that tends to $ 6, $ then, the sequence $$ \left( \sqrt[3]{s_n+\sqrt[3]{s_{n-1}+\sqrt[3]{s_{n-2}+\sqrt[3]{\cdots +\sqrt[3]{s_0}}}}} \right)_{n\ge 0} $$ tends to $ 2. $ [i]Mihai Bălună[/i]

2011 Gheorghe Vranceanu, 2
Tags: real analysis , function , lagrange , darboux , weierstrass , darboux s property
Let $ f:[0,1]\longrightarrow (0,\infty ) $ be a continuous function and $ \left( b_n \right)_{n\ge 1} $ be a sequence of numbers from the interval $ (0,1) $ that converge to $ 0. $ [b]a)[/b] Demonstrate that for any fixed $ n, $ the equation $ F(x)=b_nF(1)+\left( 1-b_n\right) F(0) $ has an unique solution, namely $ x_n, $ where $ F $ is a primitive of $ f. $ [b]b)[/b] Calculate $ \lim_{n\to\infty } \frac{x_n}{b_n} . $