$a$ is a real number. Find the all $(x,y)$ real number pairs satisfy;$$y^2=x^3+(a-1)x^2+a^2x$$$$x^2=y^3+(a-1)y^2+a^2y$$

Let $D$ be the midpoint of the side $BC$ of a triangle $ABC$ and $P$ be a point inside the $ABD$ satisfying $\angle PAD=90^\circ - \angle PBD=\angle CAD$. Prove that $\angle PQB=\angle BAC$, where $Q$ is the intersection point of the lines $PC$ and $AD$.

Given a $2015$-tuple $(a_1,a_2,\ldots,a_{2015})$ in each step we choose two indices $1\le k,l\le 2015$ with $a_k$ even and transform the $2015$-tuple into $(a_1,\ldots,\dfrac{a_k}{2},\ldots,a_l+\dfrac{a_k}{2},\ldots,a_{2015})$. Prove that starting from $(1,2,\ldots,2015)$ in finite number of steps one can reach any permutation of $(1,2,\ldots,2015)$.

Find the all $(m,n)$ integer pairs satisfying $m^4+2n^3+1=mn^3+n$.

Let $a \ge b \ge 0$ be real numbers. Find the area of the region defined as; $K=\{(x,y): x\ge y\ge0$ and $\forall n$ positive integers satisfy $a^n+b^n\ge x^n+y^n\}$ in the cordinate plane.

In a party attended by $2015$ guests among any $5$ guests at most $6$ handshakes had been exchanged. Determine the maximal possible total number of handshakes.