Find the greatest positive integer $n$ for which $n$ is divisible by all positive integers whose cube is not greater than $n.$

Let $S=\{1,2,3,\ldots,2012\}.$ We want to partition $S$ into two disjoint sets such that both sets do not contain two different numbers whose sum is a power of $2.$ Find the number of such partitions.

Let $[AB]$ be a chord of the circle $\Gamma$ not passing through its center and let $M$ be the midpoint of $[AB].$ Let $C$ be a variable point on $\Gamma$ different from $A$ and $B$ and $P$ be the point of intersection of the tangent lines at $A$ of circumcircle of $CAM$ and at $B$ of circumcircle of $CBM.$ Show that all $CP$ lines pass through a fixed point.

Find the greatest real number $M$ for which \[ a^2+b^2+c^2+3abc \geq M(ab+bc+ca) \] for all non-negative real numbers $a,b,c$ satisfying $a+b+c=4.$

Let $a, b, c$ be the side-lengths of a triangle, $r$ be the inradius and $r_a, r_b, r_c$ be the corresponding exradius. Show that \[ \frac{a+b+c}{\sqrt{a^2+b^2+c^2}} \leq 2 \cdot \frac{\sqrt{{r_a}^2+{r_b}^2+{r_c}^2}}{r_a+r_b+r_c-3r} \]

Find all positive integers $m,n$ and prime numbers $p$ for which $\frac{5^m+2^np}{5^m-2^np}$ is a perfect square.

Show that for all real numbers $x, y$ satisfying $x+y \geq 0$ \[ (x^2+y^2)^3 \geq 32(x^3+y^3)(xy-x-y) \]

Let $G$ be a connected simple graph. When we add an edge to $G$ (between two unconnected vertices), then using at most $17$ edges we can reach any vertex from any other vertex. Find the maximum number of edges to be used to reach any vertex from any other vertex in the original graph, i.e. in the graph before we add an edge.