Grades 11-12 Video Solutions 2023-2023_11-12

2023_11-12_22

Video Transcription

Problem number 22. In a bouldering competition, 13 players compete in three categories. The score  of each competitor is a product of their ranking in the three categories. For example, if one is  fourth, third, and sixth, their final score is 4 x 3 x 6 equals 72. The higher your score, the lower your overall ranking. Hannah ranks first in two of the categories. What is her  lowest possible overall ranking? Okay, so for this problem, you typically want to start with  some sort of like a guess, right? And you could say that, okay, she's probably going to be second  or third or fourth. She probably if she's already been first twice, it doesn't make sense for her  to be fifth or sixth, right? So we start with, let's say she's third. Okay, so our conjecture  is that Hannah is third. Well, if she's third, we need to prove that Hannah can't be fourth, and that being third is possible. If we can prove both those things, we'll be good to go  in solving the problem for our answer. Hannah's maximum score is equal to 13. Since the maximum  product of her scores is 1 x 1 x 13. So let's say her score is 13. Then we need three other people  who have scores lower than 13. The minimum ranking of Hannah's opponents in the first and second  category are second, third, and fourth for each. The minimum ranking in the third competition are  first, second, and third. Then the minimum product of all three competitors scores is 2 x 3 x 4 x 2 x 3 x 4 x 1 x 2 x 3.  So if we just take the product of all three scores, and that is equal to 12 x 12 x 12 x 2. However,  in order for there to be three competitors ahead of Hannah, the product of their scores  must be 12 x 12 x 12 or less. Because Hannah's number, Hannah's score was 13. So they must all  have 12 or better for them to be better. So their product must be 12 x 12 x 12 or less, which is not possible. So Hannah can't rank fourth. But can she rank third? Well, let's  show that. The competitor who ranks first could have a score of 2 x 2 x 2 equals 8. And I'm saying  they rank first in terms of the other competition. And the competitor who ranks second could have a  score of 3 x 4 x 1. This would give us 12. Then Hannah could have a final score of 13,  and her rank would be third. So exactly two people are better than her. That tells us that our correct answer is B, third place.

Video Summary

Hannah ranks first in two categories of a bouldering competition with 13 players, determining her lowest possible overall ranking. To achieve the minimum possible product of rankings (thus maximizing her score), Hannah's scores in the categories are initially assessed. With a score of 1 x 1 x 13 = 13, competitors above Hannah must have scores of 12 or less. The top-ranked competitor could have a score of 8 (ranking 2nd, 2nd, 2nd), and another could have a score of 12 (ranking 3rd, 4th, 1st). Since no three competitors can surpass her score of 13, Hannah's lowest possible ranking is 3rd.