VTMO 2015 — Grade 9 — final

Solve $\left(\sqrt{x + 24} + \sqrt{x}\right) \left(\sqrt{x + 15} - \sqrt{x + 3}\right) = 12$.

Find integer solutions with $x \gt y$ to $2 x^{3} - 9 x^{2} y + 7 xy^{2} - 3 x + 3 y - 3 = 0$.

Let $0 \lt a \lt b \lt c$ be integers with $ab + 1 = r^{2} , ac + 1 = s^{2} , bc + 1 = t^{2}$ for positive integers $r , s , t$. Prove $\frac{rs}{t}$ is not an integer.

$ABCD$ is a square with circle centered at A of radius $AB$. $M$ lies on $BC$; its non-side tangent meets $CD$ at $N$. $a$)If $MB = MC$, find $NC : ND$. $b$)Show the circumcenter of $AMN$ lies on a fixed line.

$G$ is inside triangle $ABC$; $M = AG \cap BC , N = BG \cap CA , P = CG \cap AB$. If $\frac{AG}{GM} + \frac{BG}{GN} + \frac{CG}{GP} = 6$, prove $G$ is the centroid.

$A = \{ 1 , 2 , \ldots , 100 \}$. A subset $B$ has $k$ elements, and the sum of any two distinct elements is not divisible by their difference. Find the maximum $k$.