Olympiad problems

2010 Laurențiu Panaitopol, Tulcea, 1

Tags: Sequences , limits , real analysis , Weierstrass , Nested Radicals

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]

1998 VJIMC, Problem 3

Tags: function , continuity , real analysis , limits , college contests

Give an example of a sequence of continuous functions on $\mathbb R$ converging pointwise to $0$ which is not uniformly convergent on any nonempty open set.

2011 N.N. Mihăileanu Individual, 4

Tags: real analysis , Sequences , limits

[b]a)[/b] Prove that there exists an unique sequence of real numbers $ \left( x_n \right)_{n\ge 1} $ satisfying $$ -\text{ctg} x_n=x_n\in\left( (2n+1)\pi /2,(n+1)\pi \right) , $$ for any nonnegative integer $ n. $ [b]b)[/b] Show that $ \lim_{n\to\infty } \left( \frac{x_n}{(n+1)\pi } \right)^{n^2} =e^{-1/\pi^2} . $ [i]Cătălin Zârnă[/i]

2000 Romania National Olympiad, 1

Tags: real analysis , integrals , limits , continuity , calculus , integration

Let $ a\in (1,\infty) $ and a countinuous function $ f:[0,\infty)\longrightarrow\mathbb{R} $ having the property: $$ \lim_{x\to \infty} xf(x)\in\mathbb{R} . $$ [b]a)[/b] Show that the integral $ \int_1^{\infty} \frac{f(x)}{x}dx $ and the limit $ \lim_{t\to\infty} t\int_{1}^a f\left( x^t \right) dx $ both exist, are finite and equal. [b]b)[/b] Calculate $ \lim_{t\to \infty} t\int_1^a \frac{dx}{1+x^t} . $

2003 VJIMC, Problem 3

Tags: limits , real analysis , Sequences

Let $\{a_n\}^\infty_{n=0}$ be the sequence of real numbers satisfying $a_0=0$, $a_1=1$ and $$a_{n+2}=a_{n+1}+\frac{a_n}{2^n}$$for every $n\ge0$. Prove that $$\lim_{n\to\infty}a_n=1+\sum_{n=1}^\infty\frac1{2^{\frac{n(n-1)}2}\displaystyle\prod_{k=1}^n(2^k-1)}.$$

2019 Teodor Topan, 3

Tags: Density , mod 1 , real analysis , Sequences , limits , Convergence , function

Let $ \left( c_n \right)_{n\ge 1} $ be a sequence of real numbers. Prove that the sequences $ \left( c_n\sin n \right)_{n\ge 1} ,\left( c_n\cos n \right)_{n\ge 1} $ are both convergent if and only if $ \left( c_n \right)_{n\ge 1} $ converges to $ 0. $ [i]Mihai Piticari[/i] and [i]Vladimir Cerbu[/i]

2006 Grigore Moisil Urziceni, 1

Tags: Sequences , limits

[b]a)[/b] $ \lim_{n\to\infty } \sum_{j=1}^n\frac{n}{n^2+n+j} =1 $ [b]b)[/b] $ \lim_{n\to\infty } \left( n- \sum_{j=1}^n\frac{n^2}{n^2+n+j} \right) =3/2 $ [i]Cristinel Mortici[/i]

1987 Traian Lălescu, 1.4

Tags: real analysis , Sequences , limits

[b]a)[/b] Determine all sequences of real numbers $ \left( x_n\right)_{n\in\mathbb{N}\cup\{ 0\}} $ that satisfy $ x_{n+2}+x_{n+1}=x_n, $ for any nonnegative integer $ n. $ [b]b)[/b] If $ y_k>0 $ and $ y_k^k=y_k+k, $ for all naturals $ k, $ calculate $ \lim_{n\to\infty }\frac{\ln n}{n\left( x_n-1\right)} . $

1984 Putnam, B6

Tags: limits , geometry

A sequence of convex polygons $(P_n),n\ge0,$ is defined inductively as follows. $P_0$ is an equilateral triangle with side length $1$. Once $P_n$ has been determined, its sides are trisected; the vertices of $P_{n+1}$ are the interior trisection points of the sides of $P_n$. Express $\lim_{n\to\infty}[P_n]$ in the form $\frac{\sqrt a}b$, where $a,b$ are integers.