An ant is moving on the cooridnate plane, starting form point $(0,-1)$ along a straight line until it reaches the $x$- axis at point $(x,0)$ where $x$ is a real number. After it turns $90^o$ to the left and moves again along a straight line until it reaches the $y$-axis . Then it again turns left and moves along a straight line until it reaches the $x$-axis, where it once more turns left by $90^o$ and moves along a straight line until it finally reached the $y$-axis. Can both the length of the ant's journey and distance between it's initial and final point be: (a) rational numbers ? (b) integers? Justify your answers PS. Collected [url=https://artofproblemsolving.com/community/c949609_2019_nigerian_senior_mo_round_4]here[/url]

Let a point $M$ not lying on coordinates axes be given. Points $Q$ and $P$ move along $Y$ - and $X$-axis respectively so that angle $P M Q$ is always right. Find the locus of points symmetric to $M$ wrt $P Q$.

a) What can be the number of symmetry axes of a checked polygon, that is, of a polygon whose sides lie on lines of a list of checked paper? (Indicate all possible values.) b) What can be the number of symmetry axes of a checked polyhedron, that is, of a polyhedron consisting of equal cubes which border one to another by plane facets?

A rectangle in xy Cartesian System is called latticed if all it's vertices have integer coordinates. a) Find a latticed rectangle of area $2013$, whose sides are not parallel to the axes. b) Show that if a latticed rectangle has area $2011$, then their sides are parallel to the axes.

Two straight lines intersecting at an angle of $46^o$ are the axes of symmetry of the figure $F$ on the plane. What is the smallest number of axes of symmetry this figure can have?