2011 AMC 12A Problems/Problem 22
- The following problem is from both the 2011 AMC 12A #22 and 2011 AMC 10A #25, so both problems redirect to this page.
Contents
Problem
Let be a unit square region and an integer. A point in the interior of is called n-ray partitional if there are rays emanating from that divide into triangles of equal area. How many points are -ray partitional but not -ray partitional?
Solution 1
There must be four rays emanating from that intersect the four corners of the square region. Depending on the location of , the number of rays distributed among these four triangular sectors will vary. We start by finding the corner-most point that is -ray partitional (let this point be the bottom-left-most point).
We first draw the four rays that intersect the vertices. At this point, the triangular sectors with bases as the sides of the square that the point is closest to both do not have rays dividing their areas. Therefore, their heights are equivalent since their areas are equal. The remaining rays are divided among the other two triangular sectors, each sector with rays, thus dividing these two sectors into triangles of equal areas.
Let the distance from this corner point to the closest side be and the side of the square be . From this, we get the equation . Solve for to get . Therefore, point is of the side length away from the two sides it is closest to. By moving to the right, we also move one ray from the right sector to the left sector, which determines another -ray partitional point. We can continue moving right and up to derive the set of points that are -ray partitional.
In the end, we get a square grid of points each apart from one another. Since this grid ranges from a distance of from one side to from the same side, we have a grid, a total of -ray partitional points. To find the overlap from the -ray partitional, we must find the distance from the corner-most -ray partitional point to the sides closest to it. Since the -ray partitional points form a grid, each point apart from each other, we can deduce that the -ray partitional points form a grid, each point apart from each other. To find the overlap points, we must find the common divisors of and which are and . Therefore, the overlapping points will form grids with points , , , and away from each other respectively. Since the grid with points away from each other includes the other points, we can disregard the other grids. The total overlapping set of points is a grid, which has points. Subtract from to get .
Solution 2
Position the square region so that the bottom-left corner of the square is at the origin. Then define to be the sidelength of and to be the point , where .
There must be four rays emanating from that intersect the four corners of . The areas of the four triangles formed by these rays are then , , , and .
If a point is -ray partitional, then there exist positive integers such that and . Substituting in our formulas for , , , and and canceling equal terms, we get
Taking and solving for , we get , and taking and solving for , we get . Finally, from , we have .
So for a point to be -ray partitional, , so . must then be of the form . Since is in the interior of , and can be any positive integer from to (with and just equaling and , respectively). Thus, there are points that are -ray partitional.
However, the problem asks for points that are not only -ray partitional but also not -ray partitional. Points that are -ray partitional are of the form , where and are also positive integers. We count the number of points that can also be written in this form. For a given , if and only if , and likewise with and . We can then see that a point is both -ray partitional and -ray partitional if and only if and are both divisible by . There are integers between and that are divisible by , so out of our points that are -ray partitional, are also -ray partitional.
Our answer then is just .
Solution 3
For the sake of simplicity, let be a square and set the bottom-left point as the origin. Then, has vertices:
Now, let a point in the square have coordinates
In order for the point to be ray partitional, we must be able to make triangles with area For it to not be -ray partitional, we cannot make triangles with area
When we draw such a triangle, it's base will always be on one of the sides of the square. If it is on the bottom side, then the height must be . So, the base of each triangle must be So, there will be total triangles on that side.
If the triangle is on the right side of the square, then the height has to be So, the base will be and there will be triangles.
If the triangle is on the top, the height will be , the base will be and there will be triangles along this side.
The triangles on the left will have height , base and such triangles must exist.
Simplifying, we get triangles on the respective sides of the square.
All of these numbers must be integers. Let and where
Then, the amounts become:
As long as and are integers, and will also be integers.
For this to happen, and must be of the form where So, we have a total of points that are -ray partitional.
Now, we must calculate the number of - ray partitional points that are also -ray partitional.
Using a method similar to before, if a point is -ray partitional, then we must be able to make triangles on the different sides.
So, and must be integers. This means and have to be of the form where
If a point is both -ray partitional and -ray partitional, then it can be written as . Note that whenever is divisible by , a will always exist to satisfy the above equation.
So, has to be in the range and must be divisible by . There are possibilities, namely
The -coordinate of the point and the -coordinate of the point can each use any of these possibilities, giving numbers that are both -ray partitional and -ray partitional.
Overall, we are left with solutions.
Solution 4 (not scary)
If is partitional, and there are triangles on a side, then the total area of those triangles will be . Since the side has length , then the length of the altitude from to that side will be .
There will be 1 unique way to split this side into bases of triangles with height , and also 1 unique split into triangles on the opposite side. Since can range from , there will be different distances from side, and different distances from an adjacent side, for a total of partitional points.
Similarly, for a point to be partitional, the distance from the sides should be of the form . For this point to also be partitional, it must also be of the form . Clearly, the only possible values of are for a total of 9 ways, so there are points that are partitional and partitional.
Subtracting from , we get .
-skibbysiggy
Video Solution by Ryan Yang
https://www.youtube.com/watch?v=GFJrM6Pj-Z0
See also
2011 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 21 |
Followed by Problem 23 |
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 |
2011 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 24 |
Followed by Last Problem | |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.