In triangle $ABC$, $\angle A=60^\circ$. Points $D,M$ lie on the line $AC$, and points $E,N$ lie on the line $AB$, such that $DN$ and $EM$ are the perpendicular bisectors of $AC$ and $AB$, respectively. Let $L$ be the midpoint of $MN$. Prove that $\angle EDL=\angle ELD$.
Show that any representation of $1$ as a sum of distinct reciprocals of numbers drawn from the arithmetic progression $\{2,5,8,11,\ldots\}$, such as the following example, must have at least eight terms:
$$1=\frac12+\frac15+\frac18+\frac1{11}+\frac1{20}+\frac1{41}+\frac1{110}+\frac1{1640}.$$
Consider a sequence $p_1,p_2,\ldots$ of primes such that for each $i\ge2$, either $p_i=2p_{i-1}-1$ or $p_i=2p_{i-1}+1$. An example is the sequence $2,5,11,23,47$. Show that any such sequence has a finite number of terms.
Let $n$ be a positive integer. Let $S_1,S_2,\ldots,S_k$ be a collection of $2n$-element subsets of $\{1,2,3,4,\ldots,4n-1,4n\}$ such that $S_i\cap S_j$ contains at most $n$ elements for all $1\le i\lt j\le k$. Show that
$$k\le6^{(n+1)/2}.$$