Find all integers $m$ for which the equation $$x^3 - mx^2 + mx - (m^2 + 1) = 0$$has an integer solution.

Is it possible to use $2 \times 1$ dominoes to cover a $2k \times 2k$ checkerboard which has $2$ squares, one of each colour, removed ?

Find the number of $16$-tuples $(x_1, x_2,..., x_{16})$ such that (i) $x_i = \pm 1$ for $i = 1,..., 16$, (ii) $0 \le x_1 + x_2 +... + x_r < 4$, for $r = 1, 2,... , 15$, (iii) $x_1 + x_2 +...+ x_{10} = 4$

Let $M$ and $N$ be two points on the side BC of a triangle $ABC$ such that $BM =MN = NC$. A line parallel to $AC$ meets the segments $AB, AM$ and $AN$ at the points $D, E$ and $F$ respectively. Prove that $EF = 3DE$

Find all possible values of $$ \lfloor \frac{x - p}{p} \rfloor + \lfloor \frac{-x-1}{p} \rfloor $$where $x$ is a real number and $p$ is a nonzero integer. Here $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$.

Let $f(x) = x^{1998} - x^{199}+x^{19}+ 1$. Prove that there is an infinite set of prime numbers, each dividing at least one of the integers $f(1), f(2), f(3), f(4), ...$