### Warm-up

1. Using just a compass and a ruler (without scale) divide a segment of the straight line into 4 equal segments, drawing no more then 6 lines (straight lines or circles)2. There are two villages on the same side of the river. How a road from one village to another should be built if it has to be of the smallest possible length and should touch the river?

3. In the cube two side diagonals were drawn starting at the same vertex. What is the angle between them?

4. A man went 10 miles south, then 10 miles east and finally 10 miles north. He finished at the starting point. Where are points on Earth from which one can start such journey?

5. Take two similar coins. Fix one of them. Put the other one right next to the edge of the fixed coin and mark the point /A/ on the edge of the second coin. Now roll the second coin around first one without slipping. Look carefully at the trace of the point /A/. Mark this trace with a pencil. The curve you obtained is called a cardioid. It is very often used in mechanical engineering. How many turns will the second coin make by the time it returns to its starting point?

### Sets of Points

1. Given a square 1×1, find the set of all points such that the sum of distances from each point to the sides of the square or to their extensions is equal to 4.2. There is a square

*ABCD*. Find a set of points inside of this square such that their distance to the side

*AB*is smaller then their distance to the each of the sides

*BC*,

*CD*and

*DA*.

3. Find a set of points inside a square that are more close to the center of the square, then to any of the vertices.

4. There are points

*A*,

*B*and

*C*on the plane. What is the set of points

*M*such that the distance

*MB*is neither a smallest of the distances

*MA*,

*MB*,

*MC*, nor is the greatest.