Let $m$ and $n$ be positive integers. The set $A=\{a_1,a_2,\ldots,a_m\}$ is a subset of $\{1,2,\ldots,n\}$. Whenever $a_i+a_j\le n$, $1\le i\le j\le m$, $a_i+a_j$ also belongs to $A$. Prove that
$$\frac{a_1+a_2+\cdots+a_m}{m}\ge\frac{n+1}{2}.$$
$N$ is an arbitrary point on the bisector of $\angle BAC$. $P$ and $O$ are points on the lines $AB$ and $AN$, respectively, such that $\angle ANP=90^\circ=\angle APO$. $Q$ is an arbitrary point on $NP$, and an arbitrary line through $Q$ meets the lines $AB$ and $AC$ at $E$ and $F$ respectively. Prove that $\angle OQE=90^\circ$ if and only if $QE=QF$.
For any positive integer $k$, $A_k$ is the subset of $\{k+1,k+2,\ldots,2k\}$ consisting of all elements whose digits in base $2$ contain exactly three $1$'s. Let $f(k)$ denote the number of elements in $A_k$.
(a) Prove that for any positive integer $m$, $f(k)=m$ has at least one solution.
(b) Determine all positive integers $m$ for which $f(k)=m$ has a unique solution.
Let $S$ be the set of real numbers greater than $-1$. Find all functions $f:S\to S$ such that
$$f(x+f(y)+xf(y))=y+f(x)+yf(x)$$
for all $x$ and $y$ in $S$, and $f(x)/x$ is strictly increasing for $-1\lt x\lt 0$ and for $0\lt x$.
Find a set $A$ of positive integers such that for any infinite set $P$ of prime numbers, there exist positive integers $m\in A$ and $n\notin A$, both the product of the same number (at least two) of distinct elements of $P$.