Let $\{a_n\}_{n\geq 1}$ be a sequence of real numbers which satisfies the following relation $a_{n+1}=10^n a_n^2.$ a) Prove that if $a_1$ ...

1. Let $\{a_n\}_{n\geq 1}$ be a sequence of real numbers which satisfies the following relation $a_{n+1}=10^n a_n^2.$ a) Prove that if $a_1$ is small enough, then $\displaystyle\lim_{n\to\infty} a_n =0$.
b) Find all possible values of $a_1\in \mathbb{R}$, $a_1\geq 0$, such that $\displaystyle\lim_{n\to\infty} a_n =0$.
2. There are $n$ cards numbered and stacked in increasing order from up to down (i.e. the card in the top is the number 1, the second is the 2, and so on...). With this deck, the next steps are followed:
- the first card (from the top) is put in the bottom of the deck.
- the second card (from the top) is taken away of the deck.
- the third card (from the top) is put in the bottom of the deck.
- the fourth card (from the top) is taken away of the deck.
- ...
The proccess goes on always the same way: the card in the top is put at the end of the deck and the next is taken away of the deck, until just one card is left. Determine which is that card.
3. Find all $(m,n)$ in $\mathbb{N}^2$ such that $m\mid n^2+1$ and $n\mid m^2+1$.
4. Let $\mathcal{C}$ be the family of circumferences in $\mathbb{R}^2$ that satisfy the following properties: i) if $C_n$ is the circumference with center $(n,1/2)$ and radius $1/2$, then $C_n\in \mathcal{C}$, for all $n\in \mathbb{Z}$. ii) if $C$ and $C'$, both in $\mathcal{C}$, are externally tangent, then the circunference externally tangent to $C$ and $C'$ and tanget to $x$-axis also belongs to $\mathcal{C}$. iii) $\mathcal{C}$ is the least family which these properties. Determine the set of the real numbers which are obtained as the first coordinate of the points of intersection between the elements of $\mathcal{C}$ and the $x$-axis.
5. Let $\mathbb{A}$ be the least subset of finite sequences of nonnegative integers that satisfies the following two properties
- $(0,0) \in \mathbb{A}$.
- If $(a_1,\ldots,a_n)\in \mathbb{A}$ then $$(a_1,\ldots,a_{i-2},a_{i-1}+1,1,a_{i}+1,a_{i+1},\ldots,a_n)\in \mathbb{A}$$ for all $i\in \{2,\ldots,n\}$.
For each $n\geq 2$, let $\mathbb{B}(n)$ be the set of sequences in $\mathbb{A}$ with $n$ terms. Find the number of elements of $\mathbb{B}$.
6. a) Show that if $f:[-1,1]\to \mathbb{R}$ is a convex and $C^2$ function such that $f(1),f(-1)\geq 0$, then $\min_{x\in[-1,1]} \{f(x)\} \geq - \int_{-1}^1 f''$ b) Let $B\subset \mathbb{R}^2$ the closed ball with center $0$ and radius $1$. Show that if $f: B \to \mathbb{R}$ is a convex and $C^2$ function and $f\geq 0$ in $\partial B$, then $f(0)\geq -\frac{1}{\sqrt{\pi}} \left( \int_{B} (f_{xx}f_{yy}-f_{xy}^2) \right)^{1/2}$

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