OME 1985

Let $f : P\to P$ be a bijective map from a plane $P$ to itself such that: (i) $f (r)$ is a line for every line $r$ , (ii) $f (r)$ is parallel to $r$ for every line $r$ . What possible transformations can $f$ be?

Determine if there exists a subset $E$ of $Z \times Z$ with the properties: (i) $E$ is closed under addition, (ii) $E$ contains $(0,0),$ (iii) For every $(a,b) \ne (0,0), E$ contains exactly one of $(a,b)$ and $-(a,b)$ .

Remark: We define $(a,b)+(a',b') = (a+a',b+b')$ and $-(a,b) = (-a,-b)$ .

Solve the equation $tan^2 2x+2 tan2x tan3x = 1$

Prove that for each positive integer $k$ there exists a triple $(a,b,c)$ of positive integers such that $abc = k(a+b+c)$ . In all such cases prove that $a^3+b^3+c^3$ is not a prime.

Find the equation of the circle in the complex plane determined by the roots of the equation $z^3 +(-1+i)z^2+(1-i)z+i= 0$ .

Let $OX$ and $OY$ be non-collinear rays. Through a point $A$ on $OX$ , draw two lines $r_1$ and $r_2$ that are antiparallel with respect to $\angle XOY$ . Let $r_1$ cut $OY$ at $M$ and $r_2$ cut $OY$ at $N$ . (Thus, $\angle OAM = \angle ONA$ ). The bisectors of $\angle AMY$ and $\angle ANY$ meet at $P$ . Determine the location of $P$ .