1. The transformation $ n \to 2n - 1$ or $ n \to 3n - 1$, where $ n$ is a positive integer, is called the 'change' of $ n$. Numbers $ a$ and $ b$ are called 'similar', if there exists such positive integer, that can be got by finite number of 'changes' from both $ a$ and $ b$. Find all positive integers 'similar' to $ 2005$ and less than $ 2005$.

In triangle $ ABC$ we have $ \angle{ACB} = 2\angle{ABC}$ and there exists the point $ D$ inside the triangle such that $ AD = AC$ and $ DB = DC$. Prove that $ \angle{BAC} = 3\angle{BAD}$.

Let $ x,y,z$ be positive real numbers,satisfying equality $ x^{2}+y^{2}+z^{2}=25$. Find the minimal possible value of the expression $ \frac{xy}{z} + \frac{yz}{x} + \frac{zx}{y}$.

Find all polynomials with real coefficients, for which the equality \[ P(2P(x)) = 2P(P(x)) + 2(P(x))^{2}\] holds for any real number $ x$.

Let $ ABCD$ be a convex quadrilateral. Points $ P,Q$ and $ R$ are the feets of the perpendiculars from point $ D$ to lines $ BC, CA$ and $ AB$, respectively. Prove that $ PQ=QR$ if and only if the bisectors of the angles $ ABC$ and $ ADC$ meet on segment $ AC$.

Let $ A$ be the subset of the set of positive integers, having the following $ 2$ properties: 1) If $ a$ belong to $ A$,than all of the divisors of $ a$ also belong to $ A$; 2) If $ a$ and $ b$, $ 1 < a < b$, belong to $ A$, than $ 1 + ab$ is also in $ A$; Prove that if $ A$ contains at least $ 3$ positive integers, than $ A$ contains all positive integers.

Determine all positive integers $ n$, for which $ 2^{n-1}n+1$ is a perfect square.

In a convex quadrilateral $ ABCD$ the points $ P$ and $ Q$ are chosen on the sides $ BC$ and $ CD$ respectively so that $ \angle{BAP}=\angle{DAQ}$. Prove that the line, passing through the orthocenters of triangles $ ABP$ and $ ADQ$, is perpendicular to $ AC$ if and only if the triangles $ ABP$ and $ ADQ$ have the same areas.

Let $ a_{0},a_{1},\ldots,a_{n}$ be integers, one of which is nonzero, and all of the numbers are not less than $ - 1$. Prove that if \[ a_{0} + 2a_{1} + 2^{2}a_{2} + \cdots + 2^{n}a_{n} = 0,\] then $ a_{0} + a_{1} + \cdots + a_{n} > 0$.

Let $ a,b,c$ be positive numbers, satisfying $ abc\geq 1$. Prove that \[ a^{3} + b^{3} + c^{3} \geq ab + bc + ca.\]

On the sides $ AB, BC, CD$ and $ DA$ of the rhombus $ ABCD$, respectively, are chosen points $ E, F, G$ and $ H$ so, that $ EF$ and $ GH$ touch the incircle of the rhombus. Prove that the lines $ EH$ and $ FG$ are parallel.

$ 30$ students participated in the mathematical Olympiad. Each of them was given $ 8$ problems to solve. Jury estimated their work with the following rule: 1) Each problem was worth $ k$ points, if it wasn't solved by exactly $ k$ students; 2) Each student received the maximum possible points in each problem or got $ 0$ in it; Lasha got the least number of points. What's the maximal number of points he could have? Remark: 1) means that if the problem was solved by exactly $ k$ students, than each of them got $ 30 - k$ points in it.