CIIM 2021

For every $0 < \alpha < 1$ , let $R(\alpha)$ be the region in $\mathbb{R}^2$ whose boundary is the convex pentagon of vertices $(0,1-\alpha), (\alpha, 0), (1, 0), (1,1)$ and $(0, 1)$ . Let $R$ be the set of points that belong simultaneously to each of the regions $R(\alpha)$ with $0 < \alpha < 1$ , that is, $R =\bigcap_{0<\alpha<1} R(\alpha)$ .

Determine the area of $R$ .

Let $r>s$ be positive integers. Let $P(x)$ and $Q(x)$ be distinct polynomials with real coefficients, non-constant(s), such that $P(x)^r-P(x)^s=Q(x)^r-Q(x)^s$ for every $x\in \mathbb{R}$ . Prove that $(r,s)=(2,1)$ .

Let $m,n$ and $N$ be positive integers and $\mathbb{Z}_{N}=\{0,1,\dots,N-1\}$ a set of residues modulo $N$ . Consider a table $m\times n$ such that each one of the $mn$ cells has an element of $\mathbb{Z}_{N}$ . A move is choose an element $g\in \mathbb{Z}_{N}$ , a cell in the table and add $+g$ to the elements in the same row/column of the chosen cell(the sum is modulo $N$ ). Prove that if $N$ is coprime with $m-1,n-1,m+n-1$ then any initial arrangement of your elements in the table cells can become any other arrangement using an finite quantity of moves.

Let $\mathbb{Z}^{+}$ be the set of positive integers.a) Prove that there is only one function $f:\mathbb{Z}^{+} \rightarrow \mathbb{Z}^{+}$ , strictly increasing, such that $f(f(n))=2n+1$ for every $n\in \mathbb{Z}^{+}$ .b) For the function in a. Prove that for every $n\in \mathbb{Z}^{+}$ $\frac{4n+1}{3}\leq f(n)\leq \frac{3n+1}{2}$ c) Prove that in each inequality side of b the equality can reach by infinite positive integers $n$ .

For every positive integer $n$ , let $s(n)$ be the sum of the exponents of $71$ and $97$ in the prime factorization of $n$ ; for example, $s(2021) = s(43 \cdot 47) = 0$ and $s(488977) = s(71^2 \cdot 97) = 3$ . If we define $f(n)=(-1)^{s(n)}$ , prove that the limit $$\lim_{n \to +\infty} \frac{f(1) + f(2) + \cdots+ f(n)}{n}$$ exists and determine its value.

Let $0 \le a < b$ be real numbers. Prove that there is no continuous function $f : [a, b] \to \mathbb{R}$ such that $$\int_a^b f(x)x^{2n} \mathrm dx>0 \text{and} \int_a^b f(x)x^{2n+1} \mathrm dx <0$$ for every integer $n \ge 0$ .