Some Algebra problems from IMO Shortlist that have combinatorial factors

Problem 5 of IMO 2025 is about a two-player game, but it is classified as "Algebra". Recently, IMO tends to have more and more problems from other subjects with combinatorial ideas/statements. However, there might be few known such problems which are classified as "Algebra". This post collects all such problems from IMO Shortlist. 

1. Problem A2, IMO Shortlist 1999

   The numbers 1 to n2 are arranged in the squares of an n x n board (1 per square). There are n2(n - 1) pairs of numbers in the same row or column. For each such pair take the larger number divided by the smaller. Then take the smallest such ratio and call it the minrat of the arrangement. So, for example, if n2 and n2 - 1 were in the same row, then the minrat would be n2/(n2 - 1). What is the largest possible minrat?

2. Problem A3, IMO Shortlist 1999

    Each of n > 1 girls has a doll with her name on it. Each pair of girls swaps dolls in some order. For which values of n is it possible for each girl to end up
    (1) with her own doll,
    (2) with another girl's doll?
Comment: The solution to this problem seems to require only combinatorial arguments. The key idea is to use the notion of the number of inversions of a permutation, which is important when studying determinants of matrices and permutation groups. Is it the reason why this problem is classified as "Algebra"? 

3. Problem A3, IMO Shortlist 2014

   For a sequence x_1, x_2, . . . , x_n of real numbers, we define its price as the maximal value of |x_1 + x_2 + ... + x_i| as i runs from 1 to n. Given n real numbers, Dave and George want to arrange them into a sequence with a low price. Diligent Dave checks all possible ways and finds the minimum possible price D. Greedy George, on the other hand, chooses x_1 such that |x_1| is as small as possible; among the remaining numbers, he chooses x_2 such that |x_1 + x_2| is as small as possible, and so on. Thus, in the i-th step he chooses x_i among the remaining numbers so as to minimise the value of|x_1+x_2+ ...+ x_i|. In each step, if several numbers provide the same value, George chooses one at random. Finally he gets a sequence with price G.
   Find the least possible constant c such that for every positive integer n, for every collection of n real numbers, and for every possible sequence that George might obtain, the resulting values satisfy the inequality G ≤ cD.


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