IMC 2003

(a) Let $a_1,a_2,...$ be a sequenceof reals with $a_1=1$ and $a_{n+1}>\frac32 a_n$ for all $n$ . Prove that $\lim_{n\rightarrow\infty}\frac{a_n}{\left(\frac32\right)^{n-1}}$ exists. (finite or infinite)

(b) Prove that for all $\alpha>1$ there is a sequence $a_1,a_2,...$ with the same properties such that $\lim_{n\rightarrow\infty}\frac{a_n}{\left(\frac32\right)^{n-1}}=\alpha$

Let $a_1, a_2,...,a_{51}$ be non-zero elements of a field of characteristic $p$ . We simultaneously replace each element with the sum of the 50 remaining ones. In this way we get a sequence $b_1, ... , b_{51}$ . If this new sequence is a permutation of the original one, find all possible values of $p$ .

Let $A$ be a closed subset of $\mathbb{R}^{n}$ and let $B$ be the set of all those points $b \in \mathbb{R}^{n}$ for which there exists exactly one point $a_{0}\in A$ such that $|a_{0}-b|= \inf_{a\in A}|a-b|$ . Prove that $B$ is dense in $\mathbb{R}^{n}$ ; that is, the closure of $B$ is $\mathbb{R}^{n}$