**A. 475.** To each vertex of a regular *n*-gon a real number is assigned in such a way that the sum of all *n* numbers is positive. If three consecutive vertices are assigned the numbers *x*, *y*, *z* respectively and *y*<0 then the following operation is allowed: the numbers *x*, *y*, *z *are replaced by *x*+*y*, -*y*, *z*+*y* respectively. Such an operation is performed repeatedly as long as at least one of the *n* numbers is negative. Determine whether this procedure necessarily terminates after a finite number of steps.

(5 points)

**B. 4152.** Given *n* different subsets of the set of numbers 1,2,3,...,*n*, prove that there exists a number, such that if it is left out of each subset, the remaining subsets will still be different.

(4 points)

**K. 199.** Dixie puts five-digit numbers in order, according to the following rule: First he arranges them in decreasing order of the last digit. If the last digit is the same in two numbers, the one with the smaller first digit will precede the other. If their first digits are also the same, they will be listed in decreasing order of the product of the middle three digits. (Dixie only deals with numbers whose order can be decided by this rule.) Dixie wrote six numbers on a sheet of paper. The numbers were ordered according to the rule above, but some digits got blurred. (The blurred digits are replaced by letters here.) It reads: 42348, A8318, 56B48, 8653C, 46585, D8655. Considering all possibilities, we try to guess the six numbers. (Our guess is one possible value for each number.) What is the probability that we will get all Dixie's numbers right?

(6 points)

This problem is for grade 9 students only.