TUYMAADA 2007

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 $cN$ .