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**Additional resources for Challenging mathematical problems with elementary solutions [Vol. I]**

**Example text**

Thus, for example, there are infinitely many positive integers. However, the question of determining the probability that a positive integer selected at random is divisible by 5 still makes sense; most people would say that this probability is 115, even though we have as yet given no definition applicable to this case. To formulate such a definition, consider the following more general problem. Let there be given an infinite sequence of numbers Suppose that the first N of these numbers are written on N slips of paper, the slips thoroughly mixed, and then one of them drawn at random.

On the other hand, 0 must be equidistant from D and E, and therefore must lie in the plane n which passes through the midpoint of the segment DE and is perpendicular to it. Consequently, if p intersects n, then 0 must be the point of intersection. Moreover the radius of ~ must be equal to l(OA + OD), since ~ must pass midway between the spheres ~l and ~I which have 0 as center and OA, OD respectively as radii. , if they are parallel (fig. 22), then DE is parallel to the plane ABC. In this case the only plane or sphere equidistant from A, B, C, D, E with A, B, C on one side and D, E on the other is the Fig.

For solutions see the Russian edition, Elementarnaya Geometriya Moscow, 1958, Vol. II, problem 701. SOLUTIONS 48 painted some specific color, say blue, then the total number of colorings will be five times this number. Select one of the four remaining colors, say red, and turn the cube so that the red face is the back face. This can always be done by rotating the cube about the vertical axis through its center (which leaves the blue face on top and the green face on the bottom). There are now three colors left, yellow, black, and white, with which to color the three remaining faces (the front face and the two side faces).