Difference between revisions of "2020 AMC 10B Problems/Problem 17"
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Note that there cannot be a case with <math>4</math> diameters because then there would have to be <math>5</math> diameters for the two remaining people as they have to be connected with a diameter. A contradiction arises. | Note that there cannot be a case with <math>4</math> diameters because then there would have to be <math>5</math> diameters for the two remaining people as they have to be connected with a diameter. A contradiction arises. | ||
− | Case 4: <math>5</math> | + | Case 4: <math>5</math> diameters |
There is only <math>1</math> way to do this. | There is only <math>1</math> way to do this. |
Revision as of 01:06, 17 August 2020
- The following problem is from both the 2020 AMC 10B #17 and 2020 AMC 12B #15, so both problems redirect to this page.
Contents
Problem
There are people standing equally spaced around a circle. Each person knows exactly of the other people: the people standing next to her or him, as well as the person directly across the circle. How many ways are there for the people to split up into pairs so that the members of each pair know each other?
Solution
Consider the 10 people to be standing in a circle, where two people opposite each other form a diameter of the circle.
Let us use casework on the number of diameters.
Case 1: diameters
There are ways: either pairs with , pairs with , and so on or pairs with , pairs with , etc.
Case 2: diameter
There are possible diameters to draw (everyone else pairs with the person next to them).
Note that there cannot be diameters since there would be one person on either side that will not have a pair adjacent to them. The only scenario forced is when the two people on either side would be paired up across a diameter. Thus, a contradiction will arise.
Case 3: diameters
There are possible diameters to draw.
Note that there cannot be a case with diameters because then there would have to be diameters for the two remaining people as they have to be connected with a diameter. A contradiction arises.
Case 4: diameters
There is only way to do this.
Thus, in total there are possible ways.
Video Solution
https://youtu.be/3BvJeZU3T-M?t=419 (for AMC 10) https://youtu.be/0xgTR3UEqbQ (for AMC 12)
~IceMatrix
See Also
2020 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 16 |
Followed by Problem 18 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
2020 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 14 |
Followed by Problem 16 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.