$1.$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.

Let $ABC$ be an acute-angeled triangle , non-isosceles and with barycentre $G$ (which is , in fact , the intersection of the medians).Let $M$ be the midpoint of $BC$ , and let $\Omega$ be the circle with centre $G$ and radius $GM$ , and let $N$ be the point of intersection between $\Omega$ and $BC$ that is distinct from $M$.Let $S$ be the symmetric point of $A$ with respect to $N$ , that is , the point on the line $AN$ such that $AN=NS$. Prove that $GS$ is perpendicular to $BC$.

Let $x_1,x_2, ... , x_n$ be positive integers,Asumme that in their decimal representations no $x_i$ "prolongs" $x_j$.For instance , $123$ prolongs $12$ , $459$ prolongs $4$ , but $124$ does not prolog $123$. Prove that : $\frac {1}{x_1}+\frac {1}{x_2}+...+\frac {1}{x_n} < 3$.

$4.$ Let $N$ be an integer greater than $1$.Denote by $x$ the smallest positive integer with the following property:there exists a positive integer $y$ strictly less than $x-1$ , such that $x$ divides $N+y$.Prove that x is either $p^n$ or $2p$ , where $p$ is a prime number and $n$ is a positive integer

$5.$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 $\Omega$