Find all pairs of integers $ (x,y)$, such that \[ x^2 - 2009y + 2y^2 = 0 \]

Find all real $ a$, such that there exist a function $ f: \mathbb{R}\rightarrow\mathbb{R}$ satisfying the following inequality: \[ x+af(y)\leq y+f(f(x)) \] for all $ x,y\in\mathbb{R}$

For a convex hexagon $ ABCDEF$ with an area $ S$, prove that: \[ AC\cdot(BD+BF-DF)+CE\cdot(BD+DF-BF)+AE\cdot(BF+DF-BD)\geq 2\sqrt{3}S \]

On the plane, a Cartesian coordinate system is chosen. Given points $ A_1,A_2,A_3,A_4$ on the parabola $ y = x^2$, and points $ B_1,B_2,B_3,B_4$ on the parabola $ y = 2009x^2$. Points $ A_1,A_2,A_3,A_4$ are concyclic, and points $ A_i$ and $ B_i$ have equal abscissas for each $ i = 1,2,3,4$. Prove that points $ B_1,B_2,B_3,B_4$ are also concyclic.

Given a quadrilateral $ ABCD$ with $ \angle B=\angle D=90^{\circ}$. Point $ M$ is chosen on segment $ AB$ so taht $ AD=AM$. Rays $ DM$ and $ CB$ intersect at point $ N$. Points $ H$ and $ K$ are feet of perpendiculars from points $ D$ and $ C$ to lines $ AC$ and $ AN$, respectively. Prove that $ \angle MHN=\angle MCK$.

In a checked $ 17\times 17$ table, $ n$ squares are colored in black. We call a line any of rows, columns, or any of two diagonals of the table. In one step, if at least $ 6$ of the squares in some line are black, then one can paint all the squares of this line in black. Find the minimal value of $ n$ such that for some initial arrangement of $ n$ black squares one can paint all squares of the table in black in some steps.