Let $a,b,c,d$ be real numbers such that $b-d\ge5$ and all four zeros $x_1,x_2,x_3,x_4$ of $P(x)=x^4+ax^3+bx^2+cx+d$ are real. Find the minimum of $(x_1^2+1)(x_2^2+1)(x_3^2+1)(x_4^2+1)$.
Proposed by Titu Andreescu (as credited in the supplied compilation).
Find all functions $f:\mathbb Z\to\mathbb Z$ satisfying
$$xf(2f(y)-x)+y^2f(2x-f(y))=\frac{f(x)^2}{x}+f(yf(y))$$
for every pair of integers $x,y$ with $x\ne0$.
Proposed by Titu Andreescu (as credited in the supplied compilation).
Prove that there is an infinite set of points $\{P_n:n\in\mathbb Z\}$ in the plane such that, for any three distinct integers $a,b,c$, the points $P_a,P_b,P_c$ are collinear if and only if $a+b+c=2014$.
Proposed by Sam Vandervelde (as credited in the supplied compilation).
On an infinite regular hexagonal grid, initially empty, players $A,B$ alternate, with $A$ first. On her turn $A$ puts counters in two adjacent empty cells; on his turn $B$ removes any one counter. $A$ wins as soon as $k$ consecutive cells in one line are occupied. Find the least positive $k$ for which $A$ cannot force a win in finitely many moves, or show no such minimum exists.
Proposed by Palmer Mebane (as credited in the supplied compilation).
Let $ABC$ have orthocenter $H$. Let $P$ be the second intersection of the circumcircle of $AHC$ with the internal bisector of $\angle BAC$. Let $X$ be the circumcenter of $APB$ and $Y$ the orthocenter of $APC$. Prove that $XY$ equals the circumradius of $ABC$.
Proposed by Titu Andreescu, Cosmin Pohoata (as credited in the supplied compilation).
Prove that a constant $c>0$ exists such that, whenever positive integers $a,b,n$ satisfy $\gcd(a+i,b+j)>1$ for every $i,j\in\{0,1,\ldots,n\}$, one has
$$\min\{a,b\}>(cn)^{n/2}.$$
Proposed by Gabriel Dospinescu (as credited in the supplied compilation).