Let $n$ and $k$ be positive integers. A baby uses $n^2$ blocks to form a $n\times n$ grid, with each of the blocks having a positive integer no greater than $k$ on it. The father passes by and notice that: 1. each row on the grid can be viewed as an arithmetic sequence with the left most number being its leading term, with all of them having distinct common differences; 2. each column on the grid can be viewed as an arithmetic sequence with the top most number being its leading term, with all of them having distinct common differences, Find the smallest possible value of $k$ (as a function of $n$.) Note: The common differences might not be positive. Proposed by Chu-Lan Kao

A positive integer is superb if it is the least common multiple of $1,2,\ldots, n$ for some positive integer $n$. Find all superb $x,y,z$ such that $x+y=z$. Proposed by usjl

Find all functions $f$ from real numbers to real numbers such that $$2f((x+y)^2)=f(x+y)+(f(x))^2+(4y-1)f(x)-2y+4y^2$$holds for all real numbers $x$ and $y$.

Suppose $O$ is the circumcenter of $\Delta ABC$, and $E, F$ are points on segments $CA$ and $AB$ respectively with $E, F \neq A$. Let $P$ be a point such that $PB = PF$ and $PC = PE$. Let $OP$ intersect $CA$ and $AB$ at points $Q$ and $R$ respectively. Let the line passing through $P$ and perpendicular to $EF$ intersect $CA$ and $AB$ at points $S$ and $T$ respectively. Prove that points $Q, R, S$, and $T$ are concyclic. Proposed by Li4 and usjl

Several triangles are intersecting if any two of them have non-empty intersections. Show that for any two finite collections of intersecting triangles, there exists a line that intersects all the triangles. Proposed by usjl