1. Six points are given in space such that the pairwise distances between them are all distinct. Consider the triangles with vertices at these points. Prove that the longest side in one of these triangles is at the same time the shortest side in another triangle.

2. A circle is inscribed in a given circular segment, touching its arc and chord at A and B, respectively. Prove that the line AB passes through a constant point.

3. The number xn is defined as the last digit in the decimal representation of the integer ⌊(√2)n⌋ (n =1, 2,…). Determine whether or not the sequence x1, x2,…,xn,…is periodic.

Solutions

1. Color the shortest side in each triangle black; some of the line segments may be colored more than once. If the remaining edges are colored white, then the complete graph with six vertices has all its edges colored in two colors. As is well known, any such coloring contains a triangle T whose sides are of the same color. But T must necessarily be black, because it has at least one black side—the shortest one. The longest side of T is black; it is therefore the shortest side in some other triangle.

2. Denote the circle containing the arc of the segment by c, the endpoints of the chord by M and N, and the given circle by c1. The point A is the center of a dilatation h taking c1 to c.