2015 Final tour - Azerbaijan in lower age category

$\textbf{1.}$ $a$ , $b$ , and $c$ are positive real numbers such that $abc=\frac{1}{8}$ . Prove $a^2+b^2+c^2+a^2\cdot b^2+a^2\cdot c^2+b^2\cdot c^2 \geq \frac{15}{16}$ $\textit{(2 points)}$

$\textbf{2.}$ $a$ , $b$ and $c$ are sides of a triangle. Prove that the area of triangle isn't more than $\frac{(a^2+b^2+c^2)}{6}$ . $\textit{(4 points)}$

$\textbf{3.}$ Find all $P(x)$ polynomials that have real coefficients, which for all real numbers of $x$ this equation must be true: $P(P(x))=(x^2+x+1)\cdot P(x)$ $\textit{(6 points)}$

$\textbf{4.}$ Natural number $M$ has $6$ natural divisors. If sum of this divisors is $3500$ , find all numbers $M$ . $\textit{(8 points)}$

$\textbf{5.}$ $ABCD$ is convex quadrilateral. $m\angle BAD = 90^{\circ}$ , $m\angle BAC = 2\angle BDC$ and $m\angle DBA + m\angle DBC = 180^{\circ}$ . Find $m\angle DBA$ . $\textit{(10 points)}$

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2015 Final tour - Azerbaijan in lower age category