NMC 2020

For a positive integer $n$ , denote by $g(n)$ the number of strictly ascending triples chosen from the set $\{1, 2, ..., n\}$ . Find the least positive integer $n$ such that the following holds:The number $g(n)$ can be written as the product of three different prime numbers which are (not necessarily consecutive) members in an arithmetic progression with common difference $336$ .

Georg has $2n + 1$ cards with one number written on each card. On one card the integer $0$ is written, and among the rest of the cards, the integers $k = 1, ... , n$ appear, each twice. Georg wants to place the cards in a row in such a way that the $0$ -card is in the middle, and for each $k = 1, ... , n$ , the two cards with the number $k$ have the distance $k$ (meaning that there are exactly $k - 1$ cards between them). For which $1 \le n \le 10$ is this possible?

Each of the sides $AB$ and $CD$ of a convex quadrilateral $ABCD$ is divided into three equal parts, $|AE| = |EF| = |F B|$ , $|DP| = |P Q| = |QC|$ . The diagonals of $AEPD$ and $FBCQ$ intersect at $M$ and $N$ , respectively. Prove that the sum of the areas of $\vartriangle AMD$ and $\vartriangle BNC$ is equal to the sum of the areas of $\vartriangle EPM$ and $\vartriangle FNQ$ .

Find all functions $f : R- \{-1\} \to R$ such that $$f(x)f \left( f \left(\frac{1 - y}{1 + y} \right)\right) = f\left(\frac{x + y}{xy + 1}\right)$$ for all $x, y \in R$ that satisfy $(x + 1)(y + 1)(xy + 1) \ne 0$ .