OPM 2014

The ship Meridiano do Bacalhau does its fishing business during $64$ days. Each day the capitain chooses a direction which may be either north or south and the ship sails that direction in that day. On the first day of business the ship sails $1$ mile, on the second day sails $2$ miles; generally, on the $n$ -th day it sails $n$ miles. After of the $64$ -th day, the ship was $2014$ miles north from its initial position. What is the greatest number of days that the ship could have sailed south?

Let $[ABCD]$ be a square, $M$ a point on the segment $[AD]$ , and $N$ a point on the segment $[DC]$ such that $B\hat{M}A = N\hat{M}D = 60^{\circ}$ . Calculate $M\hat{B}N$ .

Determine all natural numbers $x$ , $y$ and $z$ , such that $x\leq y\leq z$ and $$\left(1+\frac1x\right)\left(1+\frac1y\right)\left(1+\frac1z\right) = 3\text{.}$$

Let $[ABCD]$ be a convex quadrilateral with area $2014$ , and let $P$ be a point on $[AB]$ and $Q$ a point on $[AD]$ such that triangles $[ABQ]$ and $[ADP]$ have area $1$ . Let $R$ be the intersection of $[AC]$ and $[PQ]$ . Determine $\frac{\overline{RC}}{\overline{RA}}$ .