VTMO 2014 — Grade 9 — final

Solve $\sqrt{2 x + 1} + \sqrt{x} = \sqrt{6 x + 1}$.

Let $x + y + z = 5 , x^{2} + y^{2} + z^{2} = 9$. $a$)Prove $1 \le x , y , z \le \frac{7}{3}$. $b$)Find the extrema of $xyz$.

Find three integers greater than $1$ such that the product of any two plus $1$ is divisible by the third.

Circle $\left(O\right)$ has diameter $AB$. Let $C , D$ lie on it with $CD \parallel AB$. Choose $E$ on $AB$ with $AE = BD$. Extend $CE$ to meet the circle at $M$, and $DO$ to meet it at $N$. Let $F = AN \cap BM$. Prove $EF \perp AB$.

Circle $\left(O\right)$ has diameter $AB$ and $C$ lies on $AB$. Circles with diameters $AC , CB$ have tangent lengths $ME , MF$ from $M$ on $\left(O\right) , M \ne A , B$. $a$)Prove $\left(\frac{ME}{MA}\right)^{2} + \left(\frac{MF}{MB}\right)^{2} = 1$. $b$)Find $C$ such that $ME = \frac{MA}{2}$.

Students stand in a circle and shake hands with their neighbors. Let $b , g$ count boys, girls and $B , G$ count boy–boy, girl–girl handshakes. Prove $b - g = B - G$.