Prove there exist infinitely many pairs $(x,y)$ of integers $1<x<y$, such that $x^3+y \mid x+y^3$. (Dan Schwarz)

Determine all integers $n\geq 1$ for which the numbers $1,2,\ldots,n$ may be (re)ordered as $a_1,a_2,\ldots,a_n$ in such a way that the average $\dfrac {a_1+a_2+\cdots + a_k} {k}$ is an integer for all values $1\leq k\leq n$. (Dan Schwarz)

i) Show there exist (not necessarily distinct) non-negative real numbers $a_1,a_2,\ldots,a_{10}$; $b_1,b_2,\ldots,b_{10}$, with $a_k+b_k \leq 4$ for all $1\leq k \leq 10$, such that $\max\{|a_i-a_j|, |b_i-b_j|\} \geq \dfrac{4}{3} > 1$ for all $1\leq i < j \leq 10$. ii) Prove for any (not necessarily distinct) non-negative real numbers $a_1,a_2,\ldots,a_{11}$; $b_1,b_2,\ldots,b_{11}$, with $a_k+b_k \leq 4$ for all $1\leq k \leq 11$, there exist $1\leq i < j \leq 11$ such that $\max\{|a_i-a_j|, |b_i-b_j|\} \leq 1$. (Dan Schwarz)

At a point on the real line sits a greyhound. On one of the sides a hare runs, away from the hound. The only thing known is that the (maximal) speed of the hare is strictly less than the (maximal) speed of the greyhound (but not their precise ratio). Does the greyhound have a strategy for catching the hare in a finite amount of time? (Dan Schwarz)

Prove that for any integer $n>1$ there exist infinitely many pairs $(x,y)$ of integers $1<x<y$, such that $x^n+y \mid x+y^n$. (Dan Schwarz)

Let $N$ be an arbitrary positive integer. Prove that if, from among any $n$ consecutive integers larger than $N$, one may select $7$ of them, pairwise co-prime, then $n\geq 22$. (Dan Schwarz)

Let positive integers $M$, $m$, $n$ be such that $1\leq m \leq n$, $1\leq M \leq \dfrac {m(m+1)} {2}$, and let $A \subseteq \{1,2,\ldots,n\}$ with $|A|=m$. Prove there exists a subset $B\subseteq A$ with $$0 \leq \sum_{b\in B} b - M \leq n-m.$$ (Dan Schwarz)

At the junction of some countably infinite number of roads sits a greyhound. On one of the roads a hare runs, away from the junction. The only thing known is that the (maximal) speed of the hare is strictly less than the (maximal) speed of the greyhound (but not their precise ratio). Does the greyhound have a strategy for catching the hare in a finite amount of time? (Dan Schwarz)