Is it possible to place $2012$ distinct circles with the same diameter on the plane, such that each circle touches at least three others circles?

A hundred of silver coins are laid down in a line. A wizard can convert silver coin into golden one in $3$ seconds. Each golden coin, which is near the coin being converted, reduces this time by $1$ second. What minimal time is required for the wizard to convert all coins to gold?

Let $O$ be the center and $R$ be the radius of circumcircle $\omega$ of triangle $ABC$. Circle $\omega_1$ with center $O_1$ and radius $R$ pass through points $A, O$ and intersects the side $AC$ at point $K$. Let $AF$ be the diameter of circle $\omega$ and points $F, K, O_1$ are collinear. Determine $\angle ABC$:

Find all positive integers $a, b,c$ greater than $1$, such that $ab + 1$ is divisible by $c, bc + 1$ is divisible by $a$ and $ca + 1$ is divisible by $b$.

Several pupils with different heights are standing in a row. If they were arranged according to their heights, such that the highest would stand on the right, then each pupil would move for at most 8 positions. Prove that every pupil has no more than 8 pupils lower then him on his right.

Positive numbers $x, y, z$ satisfy $x + y + z \le 1$. Prove that $\big( \frac{1}{x}-1\big) \big( \frac{1}{y}-1\big)\big( \frac{1}{z}-1\big) \ge 8$.

Let $O$ be the circumcenter of triangle $ABC$: Points $D$ and $E$ are chosen at sides $AB$ and $AC$ respectively such that $\angle ADO = \angle AEO = 60^o$ and $BDEC$ is inscribed quadrangle. Prove or disprove that $ABC$ is isosceles triangle.

Finite number of dwarfs excavates ore in the mine with infinite number of levels. Each day at the same time one dwarf from each level, inhabited with exactly $n = 2, 3, ... $ dwarfs, move $n-1$ levels down. Prove that after some moment there will be no more then one dwarf on each level.

Positive numbers $x, y, z$ satisfy $x^2+y^2+z^2+xy+yz+zy \le 1$. Prove that $\big( \frac{1}{x}-1\big) \big( \frac{1}{y}-1\big)\big( \frac{1}{z}-1\big) \ge 9 \sqrt6 -19$.

Finite number of dwarfs excavates ore in the mine with infinite number of levels. Each day at the same time one dwarf from each level, inhabited with exactly $n = 1, 2, 3, ...$ dwarfs, move $n$ levels down. Prove that after some moment there will be no more then one dwarf on each level.