CIIM 2018

Show that there exists a $2 \times 2$ matrix of order 6 with rational entries, such that the sum of its entries is 2018. Note: The order of a matrix (if it exists) is the smallest positive integer $n$ such that $A^n = I$ , where $I$ is the identity matrix.

Let $p(x)$ and $q(x)$ non constant real polynomials of degree at most $n$ ( $n > 1$ ). Show that there exists a non zero polynomial $F(x,y)$ in two variables with real coefficients of degree at most $2n-2,$ such that $F(p(t),q(t)) = 0$ for every $t\in \mathbb{R}$ .

Let $m$ be an integer and $\mathbb{Z}_m$ the set of integer modulo $m$ . An equivalence relation is defined in $\mathbb{Z}_m$ given by, $x \sim y$ if there exists a natural $t$ such that $y \equiv 2^tx (\bmod m)$ . Find al values of $m$ such that the number of equivalent classes is even.

Let $\alpha < 0 < \beta$ and consider the polynomial $f(x) = x(x-\alpha)(x-\beta)$ . Let $S$ be the set of real numbers $s$ such that $f(x) - s$ has three different real roots. For $s\in S$ , let $p(x)$ the product of the smallest and largest root of $f(x)-s$ . Determine the smallest possible value that $p(s)$ for $s\in S$ .

Consider the transformation $$T(x,y,z) = (\sin y + \sin z - \sin x,\sin z + \sin x - \sin y,\sin x +\sin y -\sin z).$$ Determine all the points $(x,y,z) \in [0,1]^3$ such that $T^n(x,y,z) \in [0,1]^3,$ for every $n \geq 1$ .

Let $\{x_n\}$ be a sequence of real numbers in the interval $[0,1)$ . Prove that there exists a sequence $1 < n_1 < n_2 < n_3 < \cdots$ of positive integers such that the following limit exists $$\lim_{i,j \to \infty} x_{n_i+n_j}.$$ That is, there exists a real number $L$ such that for every $\epsilon > 0,$ there exists a positive integer $N$ such that if $i,j > N$ , then $|x_{n_i+n_j}-L| < \epsilon.$