Find all numbers $n$ that can be expressed in the form $n=k+2\lfloor\sqrt{k}\rfloor+2$ for some nonnegative integer $k$.

In trapezoid $ABCD$, the sum of the lengths of the bases $AB$ and $CD$ is equal to the length of the diagonal $BD$. Let $M$ denote the midpoint of $BC$, and let $E$ denote the reflection of $C$ about the line $DM$. Prove that $\angle AEB=\angle ACD$.

Find the largest positive integer $n$ not divisible by $10$ which is a multiple of each of the numbers obtained by deleting two consecutive digits (neither of them in the first or last position) of $n$. (Note: $n$ is written in the usual base ten notation.)

Let $P$ be a point inside triangle $ABC$ and let $R$ denote the circumradius of triangle $ABC$. Prove that \[ \frac{PA}{AB\cdot AC}+\frac{PB}{BC\cdot BA}+\frac{PC}{CA\cdot CB}\ge\frac{1}{R}.\]

Consider all finite sequences of positive real numbers each of whose terms is at most $3$ and the sum of whose terms is more than $100$. For each such sequence, let $S$ denote the sum of the subsequence whose sum is the closest to $100$, and define the defect of this sequence to be the value $|S-100|$. Find the maximum possible value of the defect.

Let $k$ be a positive integer. Show that for all $n>k$ there exist convex figures $F_{1},\ldots, F_{n}$ and $F$ such that there doesn't exist a subset of $k$ elements from $F_{1},..., F_{n}$ and $F$ is covered for this elements, but $F$ is covered for every subset of $k+1$ elements from $F_{1}, F_{2},....., F_{n}$.