A class with 2N students took a quiz, on which the possible scores were
0, 1, . . . , 10. Each score appeared at least once.
The average score for this class was exactly 7.4.
Show that the class can be divided into two groups of N members each, such that that the average score for each group was exactly 7.4.
Haven't fully thought out a solution, but it seems like you could order the students from highest score to lowest, and then split them into alternating groups so you have two lists of N scores. e.g. if the 2N scores were [
10, 10,
9, 8,
7, 7,
7, 6,
5, 5, ...] you'd end up with two lists like [
10,
9,
7,
7,
5, ...] and [10, 8, 7, 6, 5, ...]
Given the constraint that each possible score appeared at least once, when you place these lists side by side the maximum difference between any two adjacent values is 1, and the maximum difference between the sums of the two lists is 10.
And so all that's needed is to swap a pair (or more?) of students to make the sums of the two lists equal, which seems like it should always be possible. At the very least, the first list definitely contains a 10 and the second list definitely contains a 0, so you can always balance out a difference of 10 with a single swap, and smaller differences should be easy (although I haven't sketched out a proof).
|
|
Posted by tomarken
on 2024-02-24 11:20:45 |
Some thoughts
