Find all triplets $(x,y,p)$ of positive integers such that $p$ be a prime number and $\frac{xy^3}{x+y}=p$

Let $P(x)=ax^3+(b-a)x^2-(c+b)x+c$ and $Q(x)=x^4+(b-1)x^3+(a-b)x^2-(c+a)x+c$ be polynomials of $x$ with $a,b,c$ non-zero real numbers and $b>0$.If $P(x)$ has three distinct real roots $x_0,x_1,x_2$ which are also roots of $Q(x)$ then: A)Prove that $abc>28$, B)If $a,b,c$ are non-zero integers with $b>0$,find all their possible values.

Given is a triangle $ABC$ with $\angle{B}=105^{\circ}$.Let $D$ be a point on $BC$ such that $\angle{BDA}=45^{\circ}$. A) If $D$ is the midpoint of $BC$ then prove that $\angle{C}=30^{\circ}$, B) If $\angle{C}=30^{\circ}$ then prove that $D$ is the midpoint of $BC$

Square $ABCD$ with side-length $n$ is divided into $n^2$ small (fundamental) squares by drawing lines parallel to its sides (the case $n=5$ is presented on the diagram).The squares' vertices that lie inside (or on the boundary) of the triangle $ABD$ are connected with each other with arcs.Starting from $A$,we move only upwards or to the right.Each movement takes place on the segments that are defined by the fundamental squares and the arcs of the circles.How many possible roots are there in order to reach $C$;