OMCC 2021

An ordered triple $(p, q, r)$ of prime numbers is called parcera if $p$ divides $q^2-4$ , $q$ divides $r^2-4$ and $r$ divides $p^2-4$ . Find all parcera triples.

Let $ABC$ be a triangle and let $\Gamma$ be its circumcircle. Let $D$ be a point on $AB$ such that $CD$ is parallel to the line tangent to $\Gamma$ at $A$ . Let $E$ be the intersection of $CD$ with $\Gamma$ distinct from $C$ , and $F$ the intersection of $BC$ with the circumcircle of $\bigtriangleup ADC$ distinct from $C$ . Finally, let $G$ be the intersection of the line $AB$ and the internal bisector of $\angle DCF$ . Show that $E,\ G,\ F$ and $C$ lie on the same circle.

In a table consisting of $2021\times 2021$ unit squares, some unit squares are colored black in such a way that if we place a mouse in the center of any square on the table it can walk in a straight line (up, down, left or right along a column or row) and leave the table without walking on any black square (other than the initial one if it is black). What is the maximum number of squares that can be colored black?

There are $2021$ people at a meeting. It is known that one person at the meeting doesn't have any friends there and another person has only one friend there. In addition, it is true that, given any $4$ people, at least $2$ of them are friends. Show that there are $2018$ people at the meeting that are all friends with each other. *Note.*If $A$ is friend of $B$ then $B$ is a friend of $A$ .

Let $n \geq 3$ be an integer and $a_1,a_2,...,a_n$ be positive real numbers such that $m$ is the smallest and $M$ is the largest of these numbers. It is known that for any distinct integers $1 \leq i,j,k \leq n$ , if $a_i \leq a_j \leq a_k$ then $a_ia_k \leq a_j^2$ . Show that

$$a_1a_2 \cdots a_n \geq m^2M^{n-2}$$

and determine when equality holds

Let $ABC$ be a triangle with $AB<AC$ and let $M$ be the midpoint of $AC$ . A point $P$ (other than $B$ ) is chosen on the segment $BC$ in such a way that $AB=AP$ . Let $D$ be the intersection of $AC$ with the circumcircle of $\bigtriangleup ABP$ distinct from $A$ , and $E$ be the intersection of $PM$ with the circumcircle of $\bigtriangleup ABP$ distinct from $P$ . Let $K$ be the intersection of lines $AP$ and $DE$ . Let $F$ be a point on $BC$ (other than $P$ ) such that $KP=KF$ . Show that $C,\ D,\ E$ and $F$ lie on the same circle.