1

Part of 1993 Polish MO Finals

Problems(2)

System of equations in Q

Source: Problem 1, Polish NO 1993

Find all rational solutions to: \begin{eqnarray*} t^2 - w^2 + z^2 &=& 2xy \\ t^2 - y^2 + w^2 &=& 2xz \\ t^2 - w^2 + x^2 &=& 2yz . \end{eqnarray*}

algebra unsolvedalgebra

Colored convex polyhedron

Source: Problem 4, Polish NO 1993

Let be given a convex polyhedron whose all faces are triangular. The vertices of the polyhedron are colored using three colors. Prove that the number of faces with vertices in all the three colors is even.

combinatorics unsolvedcombinatorics