2022 AIME II Problems/Problem 9
Contents
Problem
Let and be two distinct parallel lines. For positive integers and , distinct points lie on , and distinct points lie on . Additionally, when segments are drawn for all and , no point strictly between and lies on more than 1 of the segments. Find the number of bounded regions into which this figure divides the plane when and . The figure shows that there are 8 regions when and
Solution 1
We can use recursion to solve this problem:
1. Fix 7 points on , then put one point on . Now, introduce a function that indicates the number of regions created, where x is the number of points on . For example, because there are 6 regions.
2. Now, put the second point on . Join and will create new regions (and we are not going to count them again), and split the existing regions. Let's focus on the spliting process: line segment formed between and intersect lines , , ..., at points creating regions (we already count one region at first), then points creating regions (we already count one region at first), 4 points, etc. So, we have:
3. If you still need one step to understand this: and will still create new regions. Intersecting at points, creating regions, etc. Thus, we have:
Yes, you might already notice that:
5. (Finally) we have , and . Therefore, the answer is .
Note: we could deduce a general formula of this recursion: , where is the number of points on
Solution 2
We want to derive a general function that indicates the number of bounded regions. Observing symmetry, we know this is a symmetric function about and . Now let's focus on , which is the difference caused by adding one point to the existing points of line . This new point, call it #m, when connected to point #1 on , crosses lines, thus making additional bounded regions; when connected to point #2 on , it crosses lines, thus making additional bounded regions; etc. By simple algebra/recursion methods, we see
Notice . Not very difficult to figure out:
The fact that makes us more confident about the formula. Now plug in , we get the final answer of .
Solution 3
Let some number of segments be constructed. We construct a new segment. We start from the straight line WLOG from point Segment will cross several existing segments (points ) and enter one of the points of the line
Each of these points adds exactly 1 new bounded region (yellow bounded regions).
The exception is the only first segment which does not create any bounded region. Thus, the number of bounded regions is less than the number of points of intersection of the segments plus the number of points of arrival of the segments to
Each point of intersection of two segments is determined uniquely by the choice of pairs of points on each line.
The number of such pairs is
Exactly one segment comes to each of the points of the line from each of the points of the line The total number of arrivals is equal to Hence, the total number of bounded regions is
We plug in , we get the final answer of .
vladimir.shelomovskii@gmail.com, vvsss
Solution 4 (Recursion with Detailed Explanation)
We use recursion for cases:
Case 1:
We add a new point on a line that already contains an equal number of points on lines and , we call the number . WLOG, let the point be on line . Then, it will have to connect to all the points on line . We observe that the number of new regions added for each line that connects the new point to some point on line is one more than the number of times the line from the new point to a point on line intersects another line formed by two arbitrary points on lines and which are not the new point. Therefore, we can focus on deriving a formula to find the number of intersections that will occur. We observe that the only way that a line chosen from two arbitrary points on lines and do not intersect with the line created from the new point to a point on line is if the line is formed by a point on line that is out of the "interval" in which the line created by the new point and the point on line go through. Therefore, if the new point connects to the furthest point on line , the number of lines that don't intersect will be the other lines formed by the point on line that connect to another point on line . Hence, the number of such lines is just , since that point on line can connect to any of the points on line which isn't the new point. Similarly, if the new point connects to the second furthest point on line , two points on line can form a line with any of the points on line . Generally, if the new point connects to the th furthest point on line , there will be lines that don't intersect the line going through the new point and the th furthest point on line . We subtract the number of lines that never intersect with the line created by the new point and a point on line from the total number of possible lines. We can do this since we know that a point will not be in the intersection of or more lines. Including all possible added regions connecting the new point to all the points on line , the equation for the total number of added regions across all possible lines can be modeled with
Case 2:
We add a point on line when there contains one more point on line then line . Following similar logic, each time that the new point connects to a point on line , the number of lines that don't intersect with the newly formed line from the new point to line is just the number of points on line that are out of the "interval" of that line multiplied by the number of points on line , which we will call if points are on line . Then, we can find the number of added regions using the equation .
For the base case , the number of regions is . Therefore, all we need to do is plug in the values for both formulas and apply for the first formula to find the number of regions when there are points on line and points on line . Next, we use the formula to find the number of regions when there are points on line and points on line after adding a point on line again (This formula can be easily derived using the same method as above). After doing some calculations, we obtain the sum
Solution 5
When a new point is added to a line, the number of newly bounded regions it creates with each line segment will be one more than the number of intersection points the line makes with other lines.
Case 1: If a new point is added to a line when both lines have an equal amount of points.
Video Solution
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See Also
2022 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 8 |
Followed by Problem 10 | |
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