IT-CESENATICO 2018

$1.$ A bottle in the shape of a cone lies on its base. Water is poured into the bottle until its level reaches a distance of 8 centimeters from the vertex of the cone (measured vertically). We now turn the bottle upside down without changing the amount of water it contains; This leaves an empty space in the upper part of the cone that is 2 centimeters high.

Find the height of the bottle.

$5.$ Let x be a real number with $0<x<1$ and let $0.c_1c_2c_3...$ be the decimal expansion of x.Denote by $B(x)$ the set of all subsequences of $c_1c_2c_3$ that consist of 6 consecutive digits. For instance , $B(\frac{1}{22})={045454,454545,545454}$ Find the minimum number of elements of $B(x)$ as $x$ varies among all irrational numbers with $0<x<1$

Let $ABC$ be a triangle with $AB=AC$ and let $I$ be its incenter. Let $\Gamma$ be the circumcircle of $ABC$ . Lines $BI$ and $CI$ intersect $\Gamma$ in two new points, $M$ and $N$ respectively. Let $D$ be another point on $\Gamma$ lying on arc $BC$ not containing $A$ , and let $E,F$ be the intersections of $AD$ with $BI$ and $CI$ , respectively. Let $P,Q$ be the intersections of $DM$ with $CI$ and of $DN$ with $BI$ respectively.

(i) Prove that $D,I,P,Q$ lie on the same circle $\Omega$ (ii) Prove that lines $CE$ and $BF$ intersect on $\Omega$