R1 contains R2, R2 contains R3, and so on. (Ri contains Ri+1 for each i. |
R1 contains R2, R2 contains R3, and so on. (Ri contains Ri+1 for each i). |
2. Given an initial sequence a1, a2, …, an of real numbers, we perform a series of steps. At each step, we replace the current sequence x1,x2,…,xn with the sequence |x1-a|,|x2-a|,…,|xn-a| for some number a that can vary with each step.
3. In a set of objects, each is either red or blue, and each is either round or square. There is at least one red object, at least one blue object, at least one round object, and at least one square object. Prove that there exist two objects that are different both in color and in shape.
4. A real number is assigned to each vertex of a finite connected graph so that the number on any vertex is the arithmetic mean (average) of the numbers on the adjacent vertices. Prove that all vertices' numbers are equal.
5. We are given an infinite set of rectangles in the plane, where each rectangle has vertices of the form (0,0), (0,n), (0,m) (n,m) for positive integers m and n (m and n vary from rectangle to rectangle).