1. Daniel is solving following problem on a heptagon table: Proove that, when a,b,c are positive rational numbers and a+ b=c, then nubers a,b are rational. So do the same!

2. In convex polygon are all inside angles equal and from any point laying inside the polygon you can see all of its sides from the same angle. Proove that this polygon is regular.

3. Proove, that every n-polygon can be divided by nonintersecting diagonals, laying inside the polygon, to triangels, and find out the dependence between the number of triangels and number n.

4. Vertices of a regular n-polygon are colored in following way. The vertices colored by color j create always another regular k_j-polygon Proove that, there are at least two equal nubers k_j.

5. We have convex n-polygon n>4. Any of its three diagonals are not intersected in one point. What is the maximum number of diagonals, that we can draw in, so the diagonals divide this n-polygon only to triangels.

The deadline for sending the solutions is 2nd november 1998.

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Solution of each problem has to be on the separate sheet of paper. Don't forget to include your name, address, school and age.

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