CIIM 2014

Let $g:[2013,2014]\to\mathbb{R}$ a function that satisfy the following two conditions:

i) $g(2013)=g(2014) = 0,$ ii) for any $a,b \in [2013,2014]$ it hold that $g\left(\frac{a+b}{2}\right) \leq g(a) + g(b).$ Prove that $g$ has zeros in any open subinterval $(c,d) \subset[2013,2014].$

Let $n$ be an integer and $p$ a prime greater than 2. Show that: $$(p-1)^nn!|(p^n-1)(p^n-p)(p^n-p^2)\cdots(p^n-p^{n-1}).$$

Given $n\geq2$ , let $\mathcal{A}$ be a family of subsets of the set $\{1,2,\dots,n\}$ such that, for any $A_1,A_2,A_3,A_4 \in \mathcal{A}$ , it holds that $|A_1 \cup A_2 \cup A_3 \cup A_4| \leq n -2$ . Prove that $|\mathcal{A}| \leq 2^{n-2}.$

Let $\{a_i\}$ be a strictly increasing sequence of positive integers. Define the sequence $\{s_k\}$ as $$s_k = \sum_{i=1}^{k}\frac{1}{[a_i,a_{i+1}]},$$ where $[a_i,a_{i+1}]$ is the least commun multiple of $a_i$ and $a_{i+1}$ . Show that the sequence $\{s_k\}$ is convergent.

A analityc function $f:\mathbb{C}\to\mathbb{C}$ is call interesting if $f(z)$ is real along the parabola $Re (z) = (Im (z))^2$ .

a) Find an example of a non constant interesting function. b) Show that every interesting function $f$ satisfy that $f'(-3/4) = 0.$

a) Let $\{x_n\}$ be a sequence with $x_n \in [0,1]$ for any $n$ . Prove that there exists $C > 0$ such that for every positive integer $r$ , there exists $m \geq 1$ and $n > m + r$ that satisfy $(n-m)|x_n-x_m| \leq C$ . b) Prove that for every $C > 0$ , there exists a sequence $\{x_n\}$ with $x_n \in [0,1]$ for all $n$ and an integer $r$ such that, if $m \geq 1$ and $n > m+r$ , then $(n-m)|x_n-x_m| > C.$