OMCPLP 2021

Juca has decided to call all positive integers with 8 digits as $sextalternados$ if it is a multiple of 30 and its consecutive digits have different parity. At the same time, Carlos decided to classify all $sextalternados$ that are multiples of 12 as $super sextalternados$ .

a) Show that $super sextalternados$ numbers don't exist.

b) Find the smallest $sextalternado$ number.

Esmeralda has created a special knight to play on quadrilateral boards that are identical to chessboards. If a knight is in a square then it can move to another square by moving 1 square in one direction and 3 squares in a perpendicular direction (which is a diagonal of a $2\times4$ rectangle instead of $2\times3$ like in chess). In this movement, it doesn't land on the squares between the beginning square and the final square it lands on.

A trip of the length $n$ of the knight is a sequence of $n$ squares $C1, C2, ..., Cn$ which are all distinct such that the knight starts at the $C1$ square and for each $i$ from $1$ to $n-1$ it can use the movement described before to go from the $Ci$ square to the $C(i+1)$ .

Determine the greatest $N \in \mathbb{N}$ such that there exists a path of the knight with length $N$ on a $5\times5$ board.

Let triangle $ABC$ be an acute triangle with $AB\neq AC$ . The bisector of $BC$ intersects the lines $AB$ and $AC$ at points $F$ and $E$ , respectively. The circumcircle of triangle $AEF$ has center $P$ and intersects the circumcircle of triangle $ABC$ at point $D$ with $D$ different to $A$ .

Prove that the line $PD$ is tangent to the circumcircle of triangle $ABC$ .

Let $x_1, x_2, x_3, x_4, x_5\in\mathbb{R}^+$ such that $$x_1^2-x_1x_2+x_2^2=x_2^2-x_2x_3+x_3^2=x_3^2-x_3x_4+x_4^2=x_4^2-x_4x_5+x_5^2=x_5^2-x_5x_1+x_1^2$$ Prove that $x_1=x_2=x_3=x_4=x_5$ .

There are 3 lines $r, s$ and $t$ on a plane. The lines $r$ and $s$ intersect perpendicularly at point $A$ . the line $t$ intersects the line $r$ at point $B$ and the line $s$ at point $C$ . There exist exactly 4 circumferences on the plane that are simultaneously tangent to all those 3 lines.

Prove that the radius of one of those circumferences is equal to the sum of the radius of the other three circumferences.

A positive integer $n$ is called $omopeiro$ if there exists $n$ non-zero integers that are not necessarily distinct such that $2021$ is the sum of the squares of those $n$ integers. For example, the number $2$ is not an $omopeiro$ , because $2021$ is not a sum of two non-zero squares, but $2021$ is an $omopeiro$ , because $2021=1^2+1^2+ \dots +1^2$ , which is a sum of $2021$ squares of the number $1$ .

Prove that there exist more than 1500 $omopeiro$ numbers.

Note: proving that there exist at least 500 $omopeiro$ numbers is worth 2 points.