OSN 2009

Find all positive integers $n\in\{1,2,3,\ldots,2009\}$ such that $$4n^6 + n^3 + 5$$ is divisible by $7$ .

For any real $x$ , let $\lfloor x\rfloor$ be the largest integer that is not more than $x$ . Given a sequence of positive integers $a_1,a_2,a_3,\ldots$ such that $a_1>1$ and $$\left\lfloor\frac{a_1+1}{a_2}\right\rfloor=\left\lfloor\frac{a_2+1}{a_3}\right\rfloor=\left\lfloor\frac{a_3+1}{a_4}\right\rfloor=\cdots$$ Prove that $$\left\lfloor\frac{a_n+1}{a_{n+1}}\right\rfloor\leq1$$ holds for every positive integer $n$ .

For every triangle $ABC$ , let $D,E,F$ be a point located on segment $BC,CA,AB$ , respectively. Let $P$ be the intersection of $AD$ and $EF$ . Prove that: $$\frac{AB}{AF}\times DC+\frac{AC}{AE}\times DB=\frac{AD}{AP}\times BC$$

In an island, there exist 7 towns and a railway system which connected some of the towns. Every railway segment connects 2 towns, and in every town there exists at least 3 railway segments that connects the town to another towns. Prove that there exists a route that visits 4 different towns once and go back to the original town. (Example: $A-B-C-D-A$ )

In a drawer, there are at most $2009$ balls, some of them are white, the rest are blue, which are randomly distributed. If two balls were taken at the same time, then the probability that the balls are both blue or both white is $\frac12$ . Determine the maximum amount of white balls in the drawer, such that the probability statement is true?

Find the lowest possible values from the function $$f(x) = x^{2008} - 2x^{2007} + 3x^{2006} - 4x^{2005} + 5x^{2004} - \cdots - 2006x^3 + 2007x^2 - 2008x + 2009$$ for any real numbers $x$ .

A pair of integers $(m,n)$ is called good if $$m\mid n^2 + n \ \text{and} \ n\mid m^2 + m$$ Given 2 positive integers $a,b > 1$ which are relatively prime, prove that there exists a good pair $(m,n)$ with $a\mid m$ and $b\mid n$ , but $a\nmid n$ and $b\nmid m$ .

Given an acute triangle $ABC$ . The incircle of triangle $ABC$ touches $BC,CA,AB$ respectively at $D,E,F$ . The angle bisector of $\angle A$ cuts $DE$ and $DF$ respectively at $K$ and $L$ . Suppose $AA_1$ is one of the altitudes of triangle $ABC$ , and $M$ be the midpoint of $BC$ . (a) Prove that $BK$ and $CL$ are perpendicular with the angle bisector of $\angle BAC$ . (b) Show that $A_1KML$ is a cyclic quadrilateral.