Let $a_1, a_2, a_3, . . .$ be a sequence of positive real numbers that satisfies $a_1 = 1$ and $a^2_{n+1} + a_{n+1} = a_n$ for every natural number $n$. Prove that $a_n \ge \frac{1}{n}$ for every natural number $n$.

Ziquan makes a drawing in the plane for art class. He starts by placing his pen at the origin, and draws a series of line segments, such that the $n^{th}$ line segment has length $n$. He is not allowed to lift his pen, so that the end of the $n^{th}$ segment is the start of the $(n + 1)^{th}$ segment. Line segments drawn are allowed to intersect and even overlap previously drawn segments. After drawing a finite number of line segments, Ziquan stops and hands in his drawing to his art teacher. He passes the course if the drawing he hands in is an $N$ by $N$ square, for some positive integer $N$, and he fails the course otherwise. Is it possible for Ziquan to pass the course?

There are $n \ge 3$ distinct positive real numbers. Show that there are at most $n-2$ different integer power of three that can be written as the sum of three distinct elements from these $n$ numbers.

$ABCD$ is a fixed rhombus. Segment $PQ$ is tangent to the inscribed circle of $ABCD$, where $P$ is on side $AB$, $Q$ is on side $AD$. Show that, when segment $PQ$ is moving, the area of $\Delta CPQ$ is a constant.

There are finite many coins in David’s purse. The values of these coins are pair wisely distinct positive integers. Is that possible to make such a purse, such that David has exactly $2020$ different ways to select the coins in his purse and the sum of these selected coins is $2020$?