Problems 1-5

Problem #1   $x$ and $y$ are positive integers. If $x+y=7$ and $x>y>0$, find $x^3-y^3+2y(y^2+x^2)+xy(x+3y)$.

Problem #2   Find a positive integer $k$ whose product of digits is equal to $\frac{11k}{4}-199$.

Problem #3   For positive integer $n$, let $f\left(n\right)$ denote the unit digit of $1+2+3+\cdots +n$. Find the value of $f\left(1\right)+f\left(2\right)+\cdots +f\left(2014\right)$. (The unit digit of $456$ is $6$, and of $759$ is $9$, etc.)

Problem #4   In pentagon $ABCDE$, $AB=BC=CD=DE$, $\angle B={96}^{\circ }$ and $\angle C=\angle D={108}^{\circ }$. Find $\angle E$.

Problem #5   Let $p$ and $q$ be positive integers such that $\frac{72}{487}<\frac{p}{q}<\frac{18}{121}$. Find the smallest possible value of $p$.