By A. M. Yaglom, I. M. Yaglom
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Extra resources for Challenging Mathematical Problems With Elementary Solutions, Vol. 1
28a. In how many ways can the number n be represented as a sum of 3 nonnegative integers x, y, z, if representations differing only in the order of the terms are not considered different? b. How many such representations are there if x, y, and z are required to be positive? * How many positive integral solutions of the equation x + y + z = n satisfy the inequalities x y + z, y x + z, z x + y? Here solutions differing only in the order of the terms are to be considered as different. ** How many incongruent triangles are there with perimeter n if the lengths of the sides are integers?
A group of 11 scientists are working on a secret project, the materials of which are kept in a safe. They want to be able to open the safe only when a majority of the group is present. Therefore the safe is provided with a number of different locks, and each scientist is given the keys to certain of these locks. How many locks are required, and how many keys must each scientist have? 10. The integers from 1 to 1000 are written in order around a circle. ). This process is continued until a number is reached which has already been marked.
N straight lines? b. n circles? ** What is the greatest number of parts into which three-dimensional space can be divided by: a. n planes? b. n spheres? * In how many points do the diagonals of a convex n-gon meet if no three diagonals intersect inside the n-gon? * Into how many parts do the diagonals of a convex n-gon divide the interior of the n-gon if no three diagonals intersect? 48. Two rectangles are considered different if they have either different dimensions or a different location. How many different rectangles consisting of an integral number of squares can be drawn a.