4/24 Problem: Let be a closed subset of . Prove that there exists a function that is zero on , positive outside , and whose partial derivatives of all orders all exist.
Solution: To be added.
5/8 Problem 1: [MS 1965/1] Let be a prime, , and a set with elements. Let be a family of partitions into nonempty parts of sizes divisible by such that the intersection of any two parts that occur in any of the partitions has at most 1 element in common. Find . (Scroll down for solution)
5/8 Problem 2: [MS 1965/2] Let be a finite commutative ring. Prove that has an identity element iff the annihilator of is 0. (The annihilator of is the set .)
Note: Here a commutative ring does not by definition need to have an identity. Multiplication is associative and commutative (i.e. is a commutative semigroup), distributive over addition, and is an additive group.
Solution: The answer is
(A) The bound is attainable: Think of as the vector space . For a line through the origin (i.e. a one-dimensional subspace of ), let the partition consist of the cosets of . The family of such partitions satisfies the given conditions: each part has elements and every two lines can intersect at only one point. The number of such partitions is just the number of lines passing through the origin. There are choices for a point other than the origin for the line to pass through, which determines the line. However, each of the points beside the origin on the line give the same line, so we must divide by .
(B) The bound is optimal. Let be the number of partitions in . Take any point . Each partition must have exactly one set containing ; call them . Since two sets in the partition cannot intersect in more than one point, are all disjoint. Each has at least elements since each set in a partition has to have at least elements. The union must have at most elements, so
giving the desired bound.