VTMO 2019 — Grade 9 — final

$a$)Solve $x^{2} + 3 x \sqrt{x + 1} = \left(\frac{27}{4}\right) \left(x + 1\right)$. $b$)Find prime pairs $\left(p , q\right)$ for which $p^{2} + 5 pq + 4 q^{2}$ is a square.

$a$)For real $a , b$ with a $^{3} + b^{3} + 6 ab = 8$, find $a + b$. $b$)Find real a such that $a + 3 \sqrt{11}$ and $\frac{1}{a} - 3 \sqrt{11}$ are integers.

Find distinct positive integers whose arithmetic and geometric means are two-digit numbers obtained by reversing the same digits.

In convex quadrilateral $ABCD , M , N$ are midpoints of $BC , AD , P = AM \cap BN , Q = DM \cap CN$. Prove $S_{PAB} + S_{QCD} = S_{MPNQ}$.

Circle $\left(O\right)$ has diameter $AB$ and $M$ is the midpoint of $OB$. The perpendicular to $AB$ at $M$ meets the circle at $C$. Choose $E$ inside $AB$; $CE$ meets the circle again at $D$. Let $BH \perp CD , H \in CD$. Prove $CH = \frac{AD}{2}$ and that the midpoint I of $DF$, where $BH$ meets the circle again at $F$, lies on a fixed circle.

Concatenate all four-digit numbers $1111$ through $9999$ in any order, producing $35556$ digits. Prove divisibility by $11$.