AZ-MO 2015

Let $a,b$ and $c$ be positive reals such that $abc=\frac{1}{8}$ .Then prove that $$a^2+b^2+c^2+a^2b^2+a^2c^2+b^2c^2\ge\frac{15}{16}$$

Let $a,b$ and $c$ be the length of sides of a triangle.Then prove that $S\le\frac{a^2+b^2+c^2}{6}$ where $S$ is the area of triangle.

Find all polynomials $P(x)$ with real coefficents such that $$P(P(x))=(x^2+x+1)\cdot P(x)$$ where $x \in \mathbb{R}$

Natural number $M$ has $6$ divisors, such that sum of them are equal to $3500$ .Find the all values of $M$ .

In the convex quadrilateral $ABCD$ angle $\angle{BAD}=90$ , $\angle{BAC}=2\cdot\angle{BDC}$ and $\angle{DBA}+\angle{DCB}=180$ . Then find the angle $\angle{DBA}$