A move on a real sequence chooses two terms and replaces each by their arithmetic mean. Show that there exists a sequence of $2015$ distinct real numbers such that, after any one initial move, a finite continuation of moves can always produce a constant sequence.
Proposed by Razvan Gelca (credited in the supplied notes).
Quadrilateral $APBQ$ is inscribed in circle $\omega$ with $\angle P = \angle Q = 90^{\circ}$ and $AP = AQ \lt BP$. Let $X$ be a variable point on segment $\overline{PQ}$. Line $AX$ meets $\omega$ again at $S$ (other than $A$). Point $T$ lies on arc $AQB$ of $\omega$ such that $\overline{XT}$ is perpendicular to $\overline{AX}$. Let $M$ denote the midpoint of chord $\overline{ST}$. As $X$ varies on segment $\overline{PQ}$, show that $M$ moves along a circle.
Proposed by Zuming Feng, Jacek Fabrykowski (credited by Evan Chen).
Find all functions $f:\mathbb Q\to\mathbb Q$ such that $f(x)+f(t)=f(y)+f(z)$ whenever rational numbers $x\lt y\lt z\lt t$ form an arithmetic progression.
Proposed by Iurie Boreico (credited in the supplied notes).
Let $ABCD$ be a cyclic quadrilateral. Prove that there is $X\in BD$ with $\angle BAC=\angle XAD$ and $\angle BCA=\angle XCD$ if and only if there is $Y\in AC$ with $\angle CBD=\angle YBA$ and $\angle CDB=\angle YDA$. Here $X,Y$ are on the respective segments.
Proposed by Sungyoon Kim (credited in the supplied notes).
Steve is piling $m\geq 1$ indistinguishable stones on the squares of an $n\times n$ grid. Each square can have an arbitrarily high pile of stones. After he finished piling his stones in some manner, he can then perform stone moves, defined as follows. Consider any four grid squares, which are corners of a rectangle, i.e. in positions $(i, k)$, $(i, l)$, $(j, k)$, $(j, l)$ for some $1\leq i, j, k, l\leq n$, such that $i\lt j$ and $k\lt l$. A stone move consists of either removing one stone from each of $(i, k)$ and $(j, l)$ and moving them to $(i, l)$ and $(j, k)$ respectively, or removing one stone from each of $(i, l)$ and $(j, k)$ and moving them to $(i, k)$ and $(j, l)$ respectively. Two ways of piling the stones are equivalent if they can be obtained from one another by a sequence of stone moves. How many different non-equivalent ways can Steve pile the stones on the grid?
Proposed by Maria Monks Gillespie (credited by Evan Chen).