OBM 2021

Let $ABC$ be a triangle with $\angle ABC=90^{\circ}$ . The square $BDEF$ is inscribed in $\triangle ABC$ , such that $D,E,F$ are in the sides $AB,CA,BC$ respectively. The inradius of $\triangle EFC$ and $\triangle EDA$ are $c$ and $b$ , respectively. Four circles $\omega_1,\omega_2,\omega_3,\omega_4$ are drawn inside the square $BDEF$ , such that the radius of $\omega_1$ and $\omega_3$ are both equal to $b$ and the radius of $\omega_2$ and $\omega_4$ are both equal to $a$ . The circle $\omega_1$ is tangent to $ED$ , the circle $\omega_3$ is tangent to $BF$ , $\omega_2$ is tangent to $EF$ and $\omega_4$ is tangent to $BD$ , each one of these circles are tangent to the two closest circles and the circles $\omega_1$ and $\omega_3$ are tangents. Determine the ratio $\frac{c}{a}$ .

A triple of positive integers $(a,b,c)$ is brazilian if $$a|bc+1$$ $$b|ac+1$$ $$c|ab+1$$ Determine all the brazilian triples.

Let $\alpha\geq 1$ be a real number. Define the set $$A(\alpha)=\{\lfloor \alpha\rfloor,\lfloor 2\alpha\rfloor, \lfloor 3\alpha\rfloor,\dots\}$$ Suppose that all the positive integers that does not belong to the $A(\alpha)$ are exactly the positive integers that have the same remainder $r$ in the division by $2021$ with $0\leq r<2021$ . Determine all the possible values of $\alpha$ .