OPM 2024

Let $ABC$ be a triangle and $D,E$ and $F$ the midpoints of sides $BC, AC$ and $BC$ . Medians $AD$ and $BE$ are perpendicular, $AD = 12$ and $BE = 9$ . What is the value of $CF$ ?

A sequence composed by $0$ s and $1$ s has at most two consecutive $0$ s. How many sequences of length $10$ exist?

A circle inscribed in the square $ABCD$ , with side $10$ cm, intersects sides $BC$ and $AD$ at points $M$ and $N$ respectively. The point $I$ is the intersection of $AM$ with the circle different from $M$ , and $P$ is the orthogonal projection of $I$ into $MN$ . Find the value of segment $PI$ .

Alexandre and Bernado are playing the following game. At the beginning, there are $n$ balls in a bag. At first turn, Alexandre can take one ball from the bag; at second turn, Bernado can take one or two balls from the bag, and so on. So they take turns and in $k$ turn, they can take a number of balls from $1$ to $k$ . Wins the one who makes the bag empty.

For each value of $n$ , find who has the winning strategy.