Difference between revisions of "2022 AIME II Problems/Problem 9"

Revision as of 16:24, 5 June 2023

Problem

Let $\ell_A$ and $\ell_B$ be two distinct parallel lines. For positive integers $m$ and $n$ , distinct points $A_1, A_2, \allowbreak A_3, \allowbreak \ldots, \allowbreak A_m$ lie on $\ell_A$ , and distinct points $B_1, B_2, B_3, \ldots, B_n$ lie on $\ell_B$ . Additionally, when segments $\overline{A_iB_j}$ are drawn for all $i=1,2,3,\ldots, m$ and $j=1,\allowbreak 2,\allowbreak 3, \ldots, \allowbreak n$ , no point strictly between $\ell_A$ and $\ell_B$ lies on more than 1 of the segments. Find the number of bounded regions into which this figure divides the plane when $m=7$ and $n=5$ . The figure shows that there are 8 regions when $m=3$ and $n=2$ $[asy] import geometry; size(10cm); draw((-2,0)--(13,0)); draw((0,4)--(10,4)); label("$\ell_A$",(-2,0),W); label("$\ell_B$",(0,4),W); point A1=(0,0),A2=(5,0),A3=(11,0),B1=(2,4),B2=(8,4),I1=extension(B1,A2,A1,B2),I2=extension(B1,A3,A1,B2),I3=extension(B1,A3,A2,B2); draw(B1--A1--B2); draw(B1--A2--B2); draw(B1--A3--B2); label("$A_1$",A1,S); label("$A_2$",A2,S); label("$A_3$",A3,S); label("$B_1$",B1,N); label("$B_2$",B2,N); label("1",centroid(A1,B1,I1)); label("2",centroid(B1,I1,I3)); label("3",centroid(B1,B2,I3)); label("4",centroid(A1,A2,I1)); label("5",(A2+I1+I2+I3)/4); label("6",centroid(B2,I2,I3)); label("7",centroid(A2,A3,I2)); label("8",centroid(A3,B2,I2)); dot(A1); dot(A2); dot(A3); dot(B1); dot(B2); [/asy]$

Solution 1

We can use recursion to solve this problem:

1. Fix 7 points on $\ell_A$ , then put one point $B_1$ on $\ell_B$ . Now, introduce a function $f(x)$ that indicates the number of regions created, where x is the number of points on $\ell_B$ . For example, $f(1) = 6$ because there are 6 regions.

2. Now, put the second point $B_2$ on $\ell_B$ . Join $A_1~A_7$ and $B_2$ will create $7$ 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 $B_2$ and $A_1$ intersect lines $\overline{B_1A_2}$ , $\overline{B_1A_3}$ , ..., $\overline{B_1A_7}$ at $6$ points $\Longrightarrow$ creating $6$ regions (we already count one region at first), then $5$ points $\Longrightarrow$ creating $5$ regions (we already count one region at first), 4 points, etc. So, we have: $f(2) = f(1) + 7 + (6+5+...+1) = 34.$

3. If you still need one step to understand this: $A_1~A_7$ and $B_3$ will still create $7$ new regions. Intersecting $\overline{A_2B_1}, \overline{A_2B_2};$ $\overline{A_3B_1}, \overline{A_3B_2};$ $...$ $\overline{A_7B_1}, \overline{A_7B_2}$ at $12$ points, creating $12$ regions, etc. Thus, we have: $f(3) = f(2)+7+(12+10+8+...+2)=34+7+6\cdot 7=83.$

Yes, you might already notice that: $f(n+1) = f(n)+7+(6+5+...+1)\cdot n = f(n) + 7 + 21n.$

5. (Finally) we have $f(4) = 153$ , and $f(5)=244$ . Therefore, the answer is $\boxed{244}$ .

Note: we could deduce a general formula of this recursion: $f(n+1)=f(n)+N_a+\frac{n\cdot (N_a) \cdot (N_a-1)}{2}$ , where $N_a$ is the number of points on $\ell_A$

Solution 2

We want to derive a general function $f(m,n)$ that indicates the number of bounded regions. Observing symmetry, we know this is a symmetric function about $m$ and $n$ . Now let's focus on $f(m+1, n)-f(m, n)$ , which is the difference caused by adding one point to the existing $m$ points of line $\ell_A$ . This new point, call it #m, when connected to point #1 on $\ell_B$ , crosses $m*(n-1)$ lines, thus making additional $m*(n-1)+1$ bounded regions; when connected to point #2 on $\ell_B$ , it crosses $m*(n-2)$ lines, thus making additional $m*(n-2)+1$ bounded regions; etc. By simple algebra/recursion methods, we see

$f(m+1, n)-f(m, n)=m*\frac{n(n-1)}{2} +n$

Notice $f(1,n)=n-1$ . Not very difficult to figure out:

$f(m, n)=\frac{m(m-1)n(n-1)}{4} +mn-1$

The fact that $f(3,2)=8$ makes us more confident about the formula. Now plug in $m=5, n=7$ , we get the final answer of $\boxed{244}$ .

Solution 3

Let some number of segments be constructed. We construct a new segment. We start from the straight line $l_B.$ WLOG from point $B_3.$ Segment will cross several existing segments (points $A,B,C,...$ ) and enter one of the points of the line $l_A (A_1).$

Each of these points adds exactly 1 new bounded region (yellow bounded regions).

The exception is the only first segment $(A_1 B_1),$ which does not create any bounded region. Thus, the number of bounded regions is $1$ less than the number of points of intersection of the segments plus the number of points of arrival of the segments to $l_A.$

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 $\dbinom{n}{2} \cdot \dbinom{m}{2}.$

Exactly one segment comes to each of the $n$ points of the line $l_A$ from each of the $m$ points of the line $l_B.$ The total number of arrivals is equal to $mn.$ Hence, the total number of bounded regions is $N = \dbinom{n}{2} \cdot \dbinom{m}{2} + mn – 1.$

We plug in $m=5, n=7$ , we get the final answer of $\boxed{244}$ .

vladimir.shelomovskii@gmail.com, vvsss

Solution 4 (Recursion with Detailed Explanation)

We use recursion for $2$ cases:

Case 1:

We add a new point on a line that already contains an equal number of points on lines $A$ and $B$ , we call the number $n$ . WLOG, let the point be on line $A$ . Then, it will have to connect to all the points on line $B$ . We observe that the number of new regions added for each line that connects the new point to some point on line $B$ is one more than the number of times the line from the new point to a point on line $B$ intersects another line formed by two arbitrary points on lines $A$ and $B$ 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 $A$ and $B$ do not intersect with the line created from the new point to a point on line $B$ is if the line is formed by a point on line $B$ that is out of the "interval" in which the line created by the new point and the point on line $B$ go through. Therefore, if the new point connects to the furthest point on line $B$ , the number of lines that don't intersect will be the other lines formed by the point on line $B$ that connect to another point on line $A$ . Hence, the number of such lines is just $n$ , since that point on line $B$ can connect to any of the $n$ points on line $A$ which isn't the new point. Similarly, if the new point connects to the second furthest point on line $B$ , two points on line $B$ can form a line with any of the $n$ points on line $A$ . Generally, if the new point connects to the $x$ th furthest point on line $B$ , there will be $x\cdot{n}$ lines that don't intersect the line going through the new point and the $n$ th furthest point on line $B$ . We subtract the number of lines that never intersect with the line created by the new point and a point on line $B$ from the total number of possible lines. We can do this since we know that a point will not be in the intersection of $3$ or more lines. Including all possible added regions connecting the new point to all the $n$ points on line $B$ , the equation for the total number of added regions across all possible lines can be modeled with $n^3-n\cdot\sum_{i=1}^{n}{i}+n\Longrightarrow n\cdot{(n^2-\sum_{i=1}^{n}{i}+1)}$

Case 2:

We add a point on line $B$ when there contains one more point on line $A$ then line $B$ . Following similar logic, each time that the new point connects to a point on line $A$ , the number of lines that don't intersect with the newly formed line from the new point to line $A$ is just the number of points on line $A$ that are out of the "interval" of that line multiplied by the number of points on line $B$ , which we will call $n$ if $n+1$ points are on line $A$ . Then, we can find the number of added regions using the equation $n\cdot{(n+1)^2}-n\cdot{\sum_{i=1}^{n+1}{i}+n+1}$ .

For the base case $n=2$ , the number of regions is $4$ . Therefore, all we need to do is plug in the values $n={3,4,5}$ for both formulas and apply $n=6$ for the first formula to find the number of regions when there are $6$ points on line $A$ and $5$ points on line $B$ . Next, we use the formula $n^2\cdot{n+1}-(n+1)\cdot\sum_{i=1}^{n}{i}+n$ to find the number of regions when there are $7$ points on line $A$ and $5$ points on line $B$ after adding a point on line $A$ again (This formula can be easily derived using the same method as above). After doing some calculations, we obtain the sum $8+9+12+22+28+45+55+65=\boxed{244}$

~Magnetoninja

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 $P$ is added to the right on a line when both lines have an equal amount of points.

WLOG, let the point be on line $\ell_A$ . We consider the complement, where new lines don't intersect other line segments. Simply observing, we see that the only line segments that don't intersect with the new lines are lines attached to some point that a new line does not pass through. Trivially, if we look at a series of points on line $\ell_B$ from left to right and a line connects $P$ to an arbitrary point, then the lines formed with that point and with remaining points on the left of that point never intersect with the line with $P$ . Let there be $s$ points on lines $\ell_A$ and $\ell_B$ before $P$ was added. For each of the $s$ points on $\ell_B$ , we subtract the total number of lines formed, which is $s^2$ , not counting $P$ . Considering all possible points on $\ell_B$ , we get $(s^2-s)+(s^2-2s)\cdots(s^2-s^2)$ total intersections. However, for each of the lines, there is one more bounded region than number of intersections, so we add $s$ . Simplifying, we get $s^3-\sum_{i=1}^{s}{i}+s\Longrightarrow s(s^2-\sum_{i=1}^{s}{i}+1)$ . Note that this is only a recursion formula to find the number of new regions added for a new point $P$ added to $\ell_A$ .

Case 2: If a new point $P$ is added to the right of a line that has one less point than the other line.

Continuing on case one, let this point $P$ be on line $\ell_B$ . With similar reasoning, we see that the idea remains the same, except $s+1$ lines are formed with $P$ instead of just $s$ lines. Once again, each line from $P$ to a point on line $\ell_A$ creates $s$ non-intersecting lines for that point and each point to its left. Subtracting from $s(s+1)$ lines and considering all possible lines created by $P$ , we get $(s(s+1)-s)+(s(s+1)-2s)\cdots(s(s+1)-s(s+1)$ intersections. However, the number of newly bounded regions is the number of intersections plus the number of points on line $\ell_A$ . Simplying, we get $s(s+1)^2-s\sum_{i=1}^{s+1}{i}+(s+1)$ newly bounded regions.

For the base case $s=2$ for both lines, there are $4$ bounded regions. Next, we plug in $s=2,3,4$ for both formulas and plug $s=5$ for the first formula to find the number of regions when $m=6$ and $n=5$ . Notice that adding a final point on $\ell_A$ is a variation of our Case 1. The only difference is for each of the $s$ lines formed by $P$ , there are $s+1$ points that can form a non-intersecting line. Therefore, we are subtracting a factor of $s+1$ lines instead of $s$ lines from a total of $s(s+1)$ lines. However, the number of lines formed by $P$ remains the same so we still add $s$ at the end when considering intersection points. Thus, the recursive equation becomes $(s(s+1)-(s+1))+(s(s+1)-2(s+1))\cdots(s(s+1)-s(s+1))+s\Longrightarrow s^2(s+1)-(s+1)\sum_{i=1}^{s}{i}+s$ . Plugging $s=5$ into this formula and adding the values we obtained from the other formulas, the final answer is $8+9+12+22+28+45+55+65=\boxed{244}$ .

Video Solution

https://youtu.be/OWNHkKlEo2A

~MathProblemSolvingSkills.com

@@ Line 110: / Line 110: @@
 Continuing on case one, let this point <math>P</math> be on line <math>\ell_B</math>. With similar reasoning, we see that the idea remains the same, except <math>s+1</math> lines are formed with <math>P</math> instead of just <math>s</math> lines. Once again, each line from <math>P</math> to a point on line <math>\ell_A</math> creates <math>s</math> non-intersecting lines for that point and each point to its left. Subtracting from <math>s(s+1)</math> lines and considering all possible lines created by <math>P</math>, we get <math>(s(s+1)-s)+(s(s+1)-2s)\cdots(s(s+1)-s(s+1)</math> intersections. However, the number of newly bounded regions is the number of intersections plus the number of points on line <math>\ell_A</math>. Simplying, we get <math>s(s+1)^2-s\sum_{i=1}^{s+1}{i}+(s+1)</math> newly bounded regions.
-For the base case <math>s=2</math> for both lines, there are <math>4</math> bounded regions. Next, we plug in <math>s=2,3,4</math> for both formulas and plug <math>s=5</math> for the first formula to find the number of regions when <math>m=6</math> and <math>n=5</math>. Notice that adding a final point on <math>\ell_A</math> is a variation of our Case 1. The only difference is for each of the <math>s</math> lines formed by <math>P</math>, there are <math>s+1</math> points that can form a non-intersecting line. Therefore, we are subtracting a factor of <math>s+1</math> lines instead of <math>s</math> lines from a total of <math>s(s+1)</math> lines. However, the number of lines formed by <math>P</math> remains the same so we still add <math>s</math> at the end when considering intersection points. Thus, the recursive equation becomes <math>(s(s+1)-(s+1))+(s(s+1)-2(s+1))\cdots(s(s+1)-s(s+1))\Longrightarrow s^2(s+1)-(s+1)\sum_{i=1}^{s}{i}+s</math>. Plugging <math>s=5</math> into this formula and adding the values we obtained from the other formulas, the final answer is <math>8+9+12+22+28+45+55+65=\boxed{244}</math>.
+For the base case <math>s=2</math> for both lines, there are <math>4</math> bounded regions. Next, we plug in <math>s=2,3,4</math> for both formulas and plug <math>s=5</math> for the first formula to find the number of regions when <math>m=6</math> and <math>n=5</math>. Notice that adding a final point on <math>\ell_A</math> is a variation of our Case 1. The only difference is for each of the <math>s</math> lines formed by <math>P</math>, there are <math>s+1</math> points that can form a non-intersecting line. Therefore, we are subtracting a factor of <math>s+1</math> lines instead of <math>s</math> lines from a total of <math>s(s+1)</math> lines. However, the number of lines formed by <math>P</math> remains the same so we still add <math>s</math> at the end when considering intersection points. Thus, the recursive equation becomes <math>(s(s+1)-(s+1))+(s(s+1)-2(s+1))\cdots(s(s+1)-s(s+1))+s\Longrightarrow s^2(s+1)-(s+1)\sum_{i=1}^{s}{i}+s</math>. Plugging <math>s=5</math> into this formula and adding the values we obtained from the other formulas, the final answer is <math>8+9+12+22+28+45+55+65=\boxed{244}</math>.
 ==Video Solution==

2022 AIME II (Problems • Answer Key • Resources)
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