CIIM 2020

Let $\alpha>1$ and consider the function $f(x)=x^{\alpha}$ for $x \ge 0$ . For $t>0$ , define $M(t)$ as the largest area that a triangle with vertices $(0, 0), (s, f(s)), (t, f(t))$ could reach, for $s \in (0,t)$ . Let $A(t)$ be the area of the region bounded by the segment with endpoints $(0, 0)$ , $(t, f(t))$ and the graph of $y =f(x)$ . (a) Show that $A(t)/M(t)$ does not depend on $t$ . We denote this value by $c(\alpha)$ . Find $c(\alpha)$ . (b) Determine the range of values of $c(\alpha)$ when $\alpha$ varies in the interval $(1, +\infty)$ .

Find all triples of positive integers $(a,b,c)$ such that the following equations are both true: I- $a^2+b^2=c^2$ II- $a^3+b^3+1=(c-1)^3$

Let $(m,r,s,t)$ be positive integers such that $m\geq s+1$ and $r\geq t$ . Consider $m$ sets $A_1, A_2, \dots, A_m$ with $r$ elements each one. Suppose that, for each $1\leq i\leq m$ , there exist at least $t$ elements of $A_i$ , such that each one(element) belongs to (at least) $s$ sets $A_j$ where $j\neq i$ . Determine the greatest quantity of elements in the following set $A_1 \cup A_2 \cup A_3 \dots \cup A_m$ .

For each polynomial $P(x)$ with real coefficients, define $P_0=P(0)$ and $P_j(x)=x^j\cdot P^{(j)}(x)$ where $P^{(j)}$ denotes the $j$ -th derivative of $P$ for $j\geq 1$ . Prove that there exists one unique sequence of real numbers $b_0, b_1, b_2, \dots$ such that for each polynomial $P(x)$ with real coefficients and for each $x$ real, we have $P(x)=b_0P_0+\sum_{k\geq 1}b_kP_k(x)=b_0P_0+b_1P_1(x)+b_2P_2(x)+\dots$

Determine all the positive real numbers $x_1, x_2, x_3, \dots, x_{2021}$ such that $x_{i+1}=\frac{x_i^3+2}{3x_i^2}$ for every $i=1, 2, 3, \dots, 2020$ and $x_{2021}=x_1$

For a set $A$ , we define $A + A = \{a + b: a, b \in A \}$ . Determine whether there exists a set $A$ of positive integers such that $$\sum_{a \in A} \frac{1}{a} = +\infty \text{and} \lim_{n \rightarrow +\infty} \frac{|(A+A) \cap \{1,2,\cdots,n \}|}{n}=0.$$ NoteGoogle translated from http://ciim.uan.edu.co/ciim-2020-pruebas-virtuales/pruebas-virtuales