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Number Theory and Combinatorics
Solution Again, the rows are the boys and the columns are the girls. We say that a row is acquainted to a column if the entry at their intersection is a positive number. All we have to do is to show that Hall’s condition is fulfilled. Choose k rows and consider the m columns acquainted to them. Color red the cells of the k rows and blue the cells of the m columns. Consequently, the cells at their intersection will be colored violet. It is not difficult to see that the entries in all red cells are zeroes. Adding up the entries of the k rows yields k, hence the entries at the violet cells add up to k as well. Adding up the entries of the m columns yields m, therefore the sum of entries in the violet and blue cells equals m. Clearly, this implies k ≤ m, so Hall’s condition is indeed fulfilled. Problem 3.109 There are b boys and g girls present at a party, where b and g are positive integers satisfying g ≥ 2b − 1. Each boy invites a girl for a dance (of course, two different boys must always invite two different girls). Prove that this can be done in such a way that every boy is either dancing with a girl he knows or all the girls he knows are not dancing. Solution If Hall’s condition is fulfilled, then each boy can invite for the dance a girl he knows. Suppose that Hall’s condition is violated and thus, we can find k boys, say b1 , b2 , . . . , bk , such that the girls they know are g1 , g2 , . . . , gm , with m < k. We choose the maximal k with this property. Now, observe that for the rest of b − k boys and g − m girls, Hall’s condition is fulfilled (otherwise the maximality of k is contradicted), hence we can make the b − k boys dance with b − k girls they know. We are left with g − m − (b − k) ≥ 2b − 1 − b + k − m ≥ k girls and we can make b1 , b2 , . . . , bk dance with k of these girls. Problem 3.110 A m × n array is filled with the numbers 1, 2, . . . , n, each used exactly m times. Show that one can always permute the numbers within columns to arrange that each row contains every number 1, 2, . . . , n exactly once. Solution Let us show first that we can permute the numbers within columns such that the first row contains every number 1, 2, . . . , n exactly once. Let the columns be the boys and let the numbers 1, 2, . . . , n be the girls. A boy (column) is acquainted with a girl (number) if that number occurs in the column. Now, consider a set of k columns; they contain km numbers, hence there exist at least k distinct numbers among them. Since Hall’s condition is fulfilled,there is a matching between the columns and the numbers 1, 2, . . . , n. Permuting these numbers to the tops of their respective columns makes the first row contain all n numbers. Finally, a simple inductive argument ends the proof. Problem 3.111 Some of the AwesomeMath students went on a trip to the beach. There were provided n buses of equal capacities for both the trip to the beach and the ride home, one student in each seat, and there were not enough seats in n − 1 buses to fit each student. Every student who left in a bus came back in a bus, but not necessarily the same one.