CIIM 2015

Find the real number $a$ such that the integral $$\int_a^{a+8}e^{-x}e^{-x^2}dx$$ attain its maximum.

Find all polynomials $P(x)$ with real coefficients that satisfy the identity $$P(x^3-2)=P(x)^3-2,$$ for every real number $x$ .

Consider the matrices $$A = \left(\begin{matrix} 1 & 2 0 & 1 \end{matrix}\right) \text{ and } B = \left(\begin{matrix} 1 & 0 2 & 1 \end{matrix}\right).$$ Let $k\geq 1$ an integer. Prove that for any nonzero $i_1,i_2,\dots,i_{k-1},j_1,j_2,\dots,j_k$ and any integers $i_0,i_k$ it holds that $$A^{i_0}B^{j_1}A^{i_1}B^{j_2}\cdots A^{i_{k-1}}B^{i_k}A^{i_k} \not = I.$$

Let $f:\mathbb{R} \to \mathbb{R}$ a continuos function and $\alpha$ a real number such that $$\lim_{x\to\infty}f(x) = \lim_{x\to-\infty}f(x) = \alpha.$$ Prove that for any $r > 0,$ there exists $x,y \in \mathbb{R}$ such that $y-x = r$ and $f(x) = f(y).$

There are $n$ people seated on a circular table that have seats numerated from 1 to $n$ clockwise. Let $k$ be a fix integer with $2 \leq k \leq n$ . The people can change their seats. There are two types of moves permitted:

1. Each person moves to the next seat clockwise.
2. Only the ones in seats 1 and $k$ exchange their seats.

Determine, in function of $n$ and $k$ , the number of possible configurations of people in the table that can be attain by using a sequence of permitted moves.

Show that there exists a real $C > 1$ that satisfy the following property: if $n > 1$ and $a_0 < a_1 < \cdots < a_n$ are positive integers such that $\frac{1}{a_0},\frac{1}{a_1},\dots,\frac{1}{a_n}$ are in arithmetic progression, then $a_0 > C^n.$