OPM 2023

Ana, Bruno and Carolina played table tennis with each other. In each game, only two of the friends played, with the third one resting. Every time one of the friends won a game, they rested during the next game. Ana played $12$ games, Bruno played $21$ games and Carolina rested for $8$ games. Who rested in the last game?

Let $[AB]$ be a diameter of a circle with center $O$ and radius $1$ . Consider $P$ a point on the circumference, different from $A$ and $B$ and let $Q$ be the midpoint of the arc $AP$ . The line parallel to $PQ$ that passes through $O$ intersects the line $PB$ at point $S$ . Determine $\overline{PS}$ .

A crate with a base of $4 \times 2$ and a height of $2$ is open at the top. Tomas wants to completely fill the crate with some of his cubes. It has $16$ equal cubes of volume $1$ and two equal cubes of volume $8$ . A cube of volume $1$ can only be placed on the top layer if the cube on the bottom layer has already been placed. In how many ways can Tom'as fill the box with cubes, placing them one by one?

Let $[ABC]$ be an equilateral triangle and $P$ be a point on $AC$ such that $\overline{PC}= 7$ . The straight line that passes through $P$ and is perpendicular to $AC$ , intersects $CB$ at point $M$ and intersects $AB$ at point $Q$ . The midpoint $N$ of $[MQ]$ is such that $\overline{BN} = 14$ . Determine the side of the triangle $[ABC]$ .

In the village of numbers the houses are numbered from $1$ to $n$ . Meanwhile, one of the houses was demolished. Duarte calculated that the average number of houses that still exist is $\frac{202}{3}$ . How many houses were there in the village and what is the number of the demolished house?

A rectangular board, where in each square there is a symbol, is said to be magnificent if, for each line $L$ and for each pair of columns $C$ and $D$ , there is on the board another line $M$ exactly equal to $L$ , except in columns $C$ and $D$ , where $M$ has symbols different from those of $L$ . What is the smallest possible number of rows on a magnificent board with $2023$ columns?