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You have $n$ rectangles of area $1$ and variable height. Pack as many of these rectangles as possible in a semicircle of area $n$. How many leftover rectangles will there be, in terms of $n$? How to ...

Consider the $d$-dimensional $\ell_1$ ball $\mathbb B_d=\{x\in\mathbb R^d: \|x\|_1\leq 1\}$, where $\|x\|_1=\sum_{i=1}^d{|x_i|}$. I'm interested in the maximum size of the (finite) subset $S\subseteq\...

Define $m(n,k,l)$ as the maximal size of a family $k$-element subsets of $[n]$ having the property that the intersection of every two sets is less than $l$. As stated on wikipedia, in 1985 Rödl ...