Positive integers $ a<b$ are given. Prove that among every $ b$ consecutive positive integers there are two numbers whose product is divisible by $ ab$.

Two polynomials $ f(x)=a_{100}x^{100}+a_{99}x^{99}+\dots+a_{1}x+a_{0}$ and $ g(x)=b_{100}x^{100}+b_{99}x^{99}+\dots+b_{1}x+b_{0}$ of degree $ 100$ differ from each other by a permutation of coefficients. It is known that $ a_{i}\ne b_{i}$ for $ i=0, 1, 2, \dots, 100$. Is it possible that $ f(x)\geq g(x)$ for all real $ x$?

$ AA_{1}$, $ BB_{1}$, $ CC_{1}$ are altitudes of an acute triangle $ ABC$. A circle passing through $ A_{1}$ and $ B_{1}$ touches the arc $ AB$ of its circumcircle at $ C_{2}$. The points $ A_{2}$, $ B_{2}$ are defined similarly. Prove that the lines $ AA_{2}$, $ BB_{2}$, $ CC_{2}$ are concurrent.

Determine maximum real $ k$ such that there exist a set $ X$ and its subsets $ Y_{1}$, $ Y_{2}$, $ ...$, $ Y_{31}$ satisfying the following conditions: (1) for every two elements of $ X$ there is an index $ i$ such that $ Y_{i}$ contains neither of these elements; (2) if any non-negative numbers $ \alpha_{i}$ are assigned to the subsets $ Y_{i}$ and $ \alpha_{1}+\dots+\alpha_{31}=1$ then there is an element $ x\in X$ such that the sum of $ \alpha_{i}$ corresponding to all the subsets $ Y_{i}$ that contain $ x$ is at least $ k$.

What minimum number of colours is sufficient to colour all positive real numbers so that every two numbers whose ratio is 4 or 8 have different colours?

Point $ D$ is chosen on the side $ AB$ of triangle $ ABC$. Point $ L$ inside the triangle $ ABC$ is such that $ BD=LD$ and $ \angle LAB=\angle LCA=\angle DCB$. It is known that $ \angle ALD+\angle ABC=180^\circ$. Prove that $ \angle BLC=90^\circ$.

Several knights are arranged on an infinite chessboard. No square is attacked by more than one knight (in particular, a square occupied by a knight can be attacked by one knight but not by two). Sasha outlined a $ 14\times 16$ rectangle. What maximum number of knights can this rectangle contain?

Prove that there exists a positive $ c$ such that for every positive integer $ N$ among any $ N$ positive integers not exceeding $ 2N$ there are two numbers whose greatest common divisor is greater than $ cN$.