CMO 2019

Let $a,b,c,d,e\geq -1$ and $a+b+c+d+e=5.$ Find the maximum and minimum value of $S=(a+b)(b+c)(c+d)(d+e)(e+a).$

Call a set of 3 positive integers $\{a,b,c\}$ a Pythagorean set if $a,b,c$ are the lengths of the 3 sides of a right-angled triangle. Prove that for any 2 Pythagorean sets $P,Q$ , there exists a positive integer $m\ge 2$ and Pythagorean sets $P_1,P_2,\ldots ,P_m$ such that $P=P_1, Q=P_m$ , and $\forall 1\le i\le m-1$ , $P_i\cap P_{i+1}\neq \emptyset$ .

Let $O$ be the circumcenter of $\triangle ABC$ ( $AB<AC$ ), and $D$ be a point on the internal angle bisector of $\angle BAC$ . Point $E$ lies on $BC$ , satisfying $OE\parallel AD$ , $DE\perp BC$ . Point $K$ lies on $EB$ extended such that $EK=EA$ . The circumcircle of $\triangle ADK$ meets $BC$ at $P\neq K$ , and meets the circumcircle of $\triangle ABC$ at $Q\neq A$ . Prove that $PQ$ is tangent to the circumcircle of $\triangle ABC$ .

Given an ellipse that is not a circle. (1) Prove that the rhombus tangent to the ellipse at all four of its sides with minimum area is unique. (2) Construct this rhombus using a compass and a straight edge.

Given is an $n\times n$ board, with an integer written in each grid. For each move, I can choose any grid, and add $1$ to all $2n-1$ numbers in its row and column. Find the largest $N(n)$ , such that for any initial choice of integers, I can make a finite number of moves so that there are at least $N(n)$ even numbers on the board.

The point $P_1, P_2,\cdots ,P_{2018}$ is placed inside or on the boundary of a given regular pentagon. Find all placement methods are made so that $$S=\sum_{1\leq i<j\leq 2018}|P_iP_j| ^2$$ takes the maximum value.