Olympiad problems

1987 Traian Lălescu, 1.4

[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)} . $

1969 Putnam, B3

Tags: Putnam , Sequences , pi

The terms of a sequence $(T_n)$ satisfy $T_n T_{n+1} =n$ for all positive integers $n$ and $$\lim_{n\to \infty} \frac{ T_{n} }{ T_{n+1}}=1.$$ Show that $ \pi T_{1}^{2}=2.$

1982 Putnam, B5

Tags: function , limits , real analysis , Sequences , analysis

For each $x>e^e$ define a sequence $S_x=u_0,u_1,\ldots$ recursively as follows: $u_0=e$, and for $n\ge0$, $u_{n+1}=\log_{u_n}x$. Prove that $S_x$ converges to a number $g(x)$ and that the function $g$ defined in this way is continuous for $x>e^e$.

2015 IMO Shortlist, N1

Tags: floor function , number theory , IMO Shortlist , Sequences

Determine all positive integers $M$ such that the sequence $a_0, a_1, a_2, \cdots$ defined by \[ a_0 = M + \frac{1}{2} \qquad \textrm{and} \qquad a_{k+1} = a_k\lfloor a_k \rfloor \quad \textrm{for} \, k = 0, 1, 2, \cdots \] contains at least one integer term.

2013 VTRMC, Problem 3

Tags: Sequences , real analysis

Define a sequence $(a_n)$ for $n\ge1$ by $a_1=2$ and $a_{n+1}=a_n^{1+n^{-3/2}}$. Is $(a_n)$ convergent (i.e. $\lim_{n\to\infty}a_n<\infty$)?