Let $m \leq n$ be two positive integers. For every $m$ and $n$, find two polynomials of degree $m$ and $n$ with $m$ distinct intersection points.
This question seems hard since we need to make a polynomial up with this property. Firstly, we can say $f(x) = a(x-x_1)(x-x_2)\cdots(x-x_n)$ and $g(x) = b(x-x'_1)(x-x_2')\cdots(x-x'_m$). Then for $f-g = c(x-x''_1)(x-x''_2)\cdots(x-x''_n)$ which we need to have $m$ real roots and as a result of the complex root conjugate theorem $n-m$ is a multiple of $2$ if we assume our polynomial to have real coefficients.