TR-MO 2014

In a bag there are $1007$ black and $1007$ white balls, which are randomly numbered $1$ to $2014$ . In every step we draw one ball and put it on the table; also if we want to, we may choose two different colored balls from the table and put them in a different bag. If we do that we earn points equal to the absolute value of their differences. How many points can we guarantee to earn after $2014$ steps?

Find all all positive integers x,y,and z satisfying the equation $x^3=3^y7^z+8$

Let $D, E, F$ be points on the sides $BC, CA, AB$ of a triangle $ABC$ , respectively such that the lines $AD, BE, CF$ are concurrent at the point $P$ . Let a line $\ell$ through $A$ intersect the rays $[DE$ and $[DF$ at the points $Q$ and $R$ , respectively. Let $M$ and $N$ be points on the rays $[DB$ and $[DC$ , respectively such that the equation $$\frac{QN^2}{DN}+\frac{RM^2}{DM}=\frac{(DQ+DR)^2-2\cdot RQ^2+2\cdot DM\cdot DN}{MN}$$ holds. Show that the lines $AD$ and $BC$ are perpendicular to each other.

Let $P$ and $Q$ be the midpoints of non-parallel chords $k_1$ and $k_2$ of a circle $\omega$ , respectively. Let the tangent lines of $\omega$ passing through the endpoints of $k_1$ intersect at $A$ and the tangent lines passing through the endpoints of $k_2$ intersect at $B$ . Let the symmetric point of the orthocenter of triangle $ABP$ with respect to the line $AB$ be $R$ and let the feet of the perpendiculars from $R$ to the lines $AP, BP, AQ, BQ$ be $R_1, R_2, R_3, R_4$ , respectively. Prove that $$\frac{AR_1}{PR_1} \cdot \frac{PR_2}{BR_2} = \frac{AR_3}{QR_3} \cdot \frac{QR_4}{BR_4}$$

Find all natural numbers $n$ for which there exist non-zero and distinct real numbers $a_1, a_2, \ldots, a_n$ satisfying $$\left\{a_i+\dfrac{(-1)^i}{a_i} \Big | 1 \leq i \leq n\right\} = \{a_i \mid 1 \leq i \leq n\}.$$

$5$ airway companies operate in a country consisting of $36$ cities. Between any pair of cities exactly one company operates two way flights. If some air company operates between cities $A, B$ and $B, C$ we say that the triple $A, B, C$ is properly-connected. Determine the largest possible value of $k$ such that no matter how these flights are arranged there are at least $k$ properly-connected triples.