In triangle $ABC$, $\angle B=90^\circ$, $AB>BC$, and $P$ is the point such that $BP=BC$ and $\angle APB=90^\circ$, where $P$ and $C$ lie on the same side of $AB$. Let $Q$ be the point on $AB$ such that $AP=AQ$, and let $M$ be the midpoint of $QC$. Prove that the line through $M$ parallel to $AP$ passes through the midpoint of $AB$.
Let $n$ be a fixed positive integer. Find the minimum value of
$$\frac{x_1^3+\cdots+x_n^3}{x_1+\cdots+x_n},$$
where $x_1,x_2,\ldots,x_n$ are distinct positive integers.
Alice and Bob play a game. Bob starts by picking a set $S$ consisting of $M$ vectors of length $n$ with entries either $0$ or $1$. Alice picks a sequence of numbers $y_1\le y_2\le\cdots\le y_n$ from the interval $[0,1]$, and a choice of real numbers $x_1,x_2,\ldots,x_n\in\mathbb R$. Bob wins if he can pick a vector $(z_1,z_2,\ldots,z_n)\in S$ such that
$$\sum_{i=1}^n x_i y_i\le\sum_{i=1}^n x_i z_i,$$
otherwise Alice wins. Determine the minimum value of $M$ so that Bob can guarantee a win.
Let $p$ be a prime number. Determine the largest possible $n$ such that the following holds: it is possible to fill an $n\times n$ table with integers $a_{ik}$ in the $i$th row and $k$th column, for $1\le i,k\le n$, such that for any quadruple $i,j,k,l$ with $1\le i\lt j\le n$ and $1\le k\lt l\le n$, the number $a_{ik}a_{jl}-a_{il}a_{jk}$ is not divisible by $p$.