UZ-MO 2012

Given a digits { $0,1,2,...,9$ } . Find the number of numbers of 6 digits which cantain $7$ or $7$ 's digit and they is permulated(For example 137456 and 314756 is one numbers).

For any positive integers $n$ and $m$ satisfying the equation $n^3+(n+1)^3+(n+2)^3=m^3$ , prove that $4\mid n+1$ .

Given $a,b$ and $c$ positive real numbers with $ab+bc+ca=1$ . Then prove that $\frac{a^3}{1+9b^2ac}+\frac{b^3}{1+9c^2ab}+\frac{c^3}{1+9a^2bc} \geq \frac{(a+b+c)^3}{18}$

Given points $A,B,C$ and $D$ lie a circle. $AC\cap BD=K$ . $I_1, I_2,I_3$ and $I_4$ incenters of $ABK,BCK,CDK,DKA$ . $M_1,M_2,M_3,M_4$ midpoints of arcs $AB,BC,CA,DA$ . Then prove that $M_1I_1,M_2I_2,M_3I_3,M_4I_4$ are concurrent.