Difference between revisions of "2021 AMC 12B Problems/Problem 25"

Revision as of 19:17, 16 July 2024

The following problem is from both the 2021 AMC 10B #25 and 2021 AMC 12B #25, so both problems redirect to this page.

Problem

Let $S$ be the set of lattice points in the coordinate plane, both of whose coordinates are integers between $1$ and $30,$ inclusive. Exactly $300$ points in $S$ lie on or below a line with equation $y=mx.$ The possible values of $m$ lie in an interval of length $\frac ab,$ where $a$ and $b$ are relatively prime positive integers. What is $a+b?$

$\textbf{(A)} ~31 \qquad \textbf{(B)} ~47 \qquad \textbf{(C)} ~62\qquad \textbf{(D)} ~72 \qquad \textbf{(E)} ~85$

Solution 1

First, we find a numerical representation for the number of lattice points in $S$ that are under the line $y=mx.$ For any value of $x,$ the highest lattice point under $y=mx$ is $\lfloor mx \rfloor.$ Because every lattice point from $(x, 1)$ to $(x, \lfloor mx \rfloor)$ is under the line, the total number of lattice points under the line is $\sum_{x=1}^{30}(\lfloor mx \rfloor).$

Now, we proceed by finding lower and upper bounds for $m.$ To find the lower bound, we start with an approximation. If $300$ lattice points are below the line, then around $\frac{1}{3}$ of the area formed by $S$ is under the line. By using the formula for a triangle's area, we find that when $x=30, y \approx 20.$ Solving for $m$ assuming that $(30, 20)$ is a point on the line, we get $m = \frac{2}{3}.$ Plugging in $m$ to $\sum_{x=1}^{30}(\lfloor mx \rfloor),$ we get:

$\sum_{x=1}^{30}(\lfloor \frac{2}{3}x \rfloor) = 0 + 1 + 2 + 2 + 3 + \cdots + 18 + 18 + 19 + 20$

We have a repeat every $3$ values (every time $y=\frac{2}{3}x$ goes through a lattice point). Thus, we can use arithmetic sequences to calculate the value above:

$\sum_{x=1}^{30}(\lfloor \frac{2}{3}x \rfloor) = 0 + 1 + 2 + 2 + 3 + \cdots + 18 + 18 + 19 + 20$ $=\frac{20(21)}{2} + 2 + 4 + 6 + \cdots + 18$ $=210 + \frac{20}{2}\cdot 9$ $=300$

This means that $\frac{2}{3}$ is a possible value of $m.$ Furthermore, it is the lower bound for $m.$ This is because $y=\frac{2}{3}x$ goes through many points (such as $(21, 14)$ ). If $m$ was lower, $y=mx$ would no longer go through some of these points, and there would be less than $300$ lattice points under it.

Now, we find an upper bound for $m.$ Imagine increasing $m$ slowly and rotating the line $y=mx,$ starting from the lower bound of $m=\frac{2}{3}.$ The upper bound for $m$ occurs when $y=mx$ intersects a lattice point again (look at this problem to get a better idea of what's happening: https://artofproblemsolving.com/wiki/index.php/2011_AMC_10B_Problems/Problem_24).

In other words, we are looking for the first $m > \frac{2}{3}$ that is expressible as a ratio of positive integers $\frac{p}{q}$ with $q \le 30.$ For each $q=1,\dots,30$ , the smallest multiple of $\frac{1}{q}$ which exceeds $\frac{2}{3}$ is $1, \frac{2}{2}, \frac{3}{3}, \frac{3}{4}, \frac{4}{5}, \cdots , \frac{19}{27}, \frac{19}{28}, \frac{20}{29}, \frac{21}{30}$ respectively, and the smallest of these is $\frac{19}{28}.$

Alternatively, see the method of finding upper bounds in solution 2.

The lower bound is $\frac{2}{3}$ and the upper bound is $\frac{19}{28}.$ Their difference is $\frac{1}{84},$ so the answer is $1 + 84 = \boxed{85}.$

~JimY ~Minor edits by sl_hc

An alternative would be using Farey fractions and the mediant theorem to find the upper bound. $\frac{2}{3}$ and $\frac{7}{10}$ gives $\frac{9}{13},$ and so on using Farey addition.

An alternative approach with the same methodology can be done using Pick's Theorem. Wikipedia page: https://en.wikipedia.org/wiki/Pick%27s_theorem. It's a formula to find the number of lattice points strictly inside a polygon. Approximation of the lower bound is still necessary.

Solution 2

I know that I want about $\frac{2}{3}$ of the box of integer coordinates above my line. There are a total of $30$ integer coordinates in the desired range for each axis which gives a total of $900$ lattice points. I estimate that the slope, $m$ , is $\frac{2}{3}$ . Now, although there is probably an easier solution, I would try to count the number of points above the line to see if there are $600$ points above the line. The line $y=\frac{2}{3}x$ separates the area inside the box so that $\frac{2}{3}$ of the area is above the line.

I find that the number of coordinates with $x=1$ above the line is $30$ , and the number of coordinates with $x=2$ above the line is 29. Every time the line $y=\frac{2}{3}x$ hits a y-value with an integer coordinate, the number of points above the line decreases by one. I wrote out the sum of $30$ terms in hopes of finding a pattern. I graphed the first couple positive integer x-coordinates, and found that the sum of the integers above the line is $30+29+28+28+27+26+26 \ldots+ 10$ . The even integer repeats itself every third term in the sum. I found that the average of each of the terms is $20$ , and there are $30$ of them which means that exactly $600$ above the line as desired. This give a lower bound because if the slope decreases a little bit, then the points that the line goes through will be above the line.

To find the upper bound, notice that each point with an integer-valued x-coordinate is either $\frac{1}{3}$ or $\frac{2}{3}$ above the line. Since the slope through a point is the y-coordinate divided by the x-coordinate, a shift in the slope will increase the y-value of the higher x-coordinates. We turn our attention to $x=28, 29, 30$ which the line $y=\frac{2}{3}x$ intersects at $y= \frac{56}{3}, \frac{58}{3}, 20$ . The point $(30,20)$ is already counted below the line, and we can clearly see that if we slowly increase the slope of the line, we will hit the point $(28,19)$ since $(28, \frac{56}{3})$ is closer to the lattice point. The equation of the line which goes through both the origin and $(28,19)$ is $y=\frac{19}{28}x$ . This gives an upper bound of $m=\frac{19}{28}$ .

Taking the upper bound of m and subtracting the lower bound yields $\frac{19}{28}-\frac{2}{3}=\frac{1}{84}$ . This is answer $1+84=$ $\boxed{\textbf{(E)} ~85}$ .

~theAJL

Diagram

$[asy] /* Created by Brendanb4321 */ import graph; size(16cm); defaultpen(fontsize(9pt)); xaxis(0,30,Ticks(1.0)); yaxis(0,25,Ticks(1.0)); draw((0,0)--(30,20)); draw((0,0)--(30,30/28*19), dotted); for (int i = 1; i<=30; ++i) { for (int j = 1; j<=2/3*i+1; ++j) { dot((i,j)); } } dot((28,19), red); label("$m=2/3$", (32,20)); label("$m=19/28$", (32.3,20.8)); [/asy]$

Solution 3

As the procedure shown in the Solution 1, the lower bound of $m$ is $\frac{2}{3}.$ Here I give a more logic way to show how to find the upper bound of $m.$ Denote $N=\sum_{x=1}^{30}(\lfloor mx \rfloor)$ as the number of lattice points in $S$ .

$N = \lfloor m \rfloor+\lfloor 2m \rfloor+\lfloor 3m \rfloor+\cdots+\lfloor 30m \rfloor = 300 .$

Let $m = \frac{2}{3}+k$ . for $\forall x_{i}\le 30, x\in N^{*}, \lfloor mx_{i} \rfloor = \lfloor \frac{2}{3}x+xk \rfloor.$

Our target is finding the minimum value of $k$ which can increase one unit of $\lfloor mx_{i} \rfloor .$

Notice that:

When $x_{i} = 3n, \lfloor mx_{i} \rfloor = \lfloor 2n+(3n)k \rfloor$ We don't have to discuss this case.

When $x_{i} = 3n+1, \lfloor mx_{i} \rfloor = \lfloor 2n+\frac{2}{3}+(3n+1)k \rfloor, k_{min1}=\frac{1}{3(3n+1)}.$

When $x_{i} = 3n+2, \lfloor mx_{i} \rfloor = \lfloor 2n+1+\frac{1}{3}+(3n+2)k \rfloor, k_{min2}=\frac{2}{3(3n+2)}.$

Here $n\in N^{*}, n \le 9.$

Denote $k_{min}=min\left \{k_{min1},k_{min2} \right \}.$

Obviously $k_{min1}$ and $k_{min2}$ are decreasing. Let's considering the situation when $n=9.$

$k_{min}=min\left\{\frac{1}{84},\frac{2}{87}\right\}=\frac{1}{84}.$

This means that the answer is just $\frac{1}{84}$ , so $a+b=85$ . Choose $\boxed{E}.$

~PythZhou.

Solution 4

It's easier to calculate the number of lattice points inside a rectangle with vertices $(0,0)$ , $(p,0)$ , $(p,q)$ , $(0,q)$ . Those lattice points are divided by the diagonal $y = \frac{p}{q} \cdot x$ into $2$ halves. In this problem the number of lattice points on or below the diagonal and $x \ge 1$ is

,  is the number of lattice points on the diagonal,

$(p+1)(q+1)$ is the total number of lattice points inside the rectangle. Subtract the number of lattice points on the diagonal, divided by 2 is the number of lattice points below the diagonal, add the number of lattice points on the diagonal, and subtract the lattice points on the $x$ axis, then we get the total number of lattice points on or below the diagonal and $x \ge 1$ .

There are $900$ lattice points in total. $300$ is $\frac{1}{3}$ of $900$ . The $x$ coordinate of the top-right vertex of the rectangle is $30$ , $\frac{1}{2} \cdot 30 \cdot 20 = 300$ . I guess the $y$ coordinate of the top-right vertex of the rectangle is $20$ . Now I am going to verify that. The slope of the diagonal is $\frac{20}{30} = \frac{2}{3}$ , there are $11$ lattice points on the diagonal. Substitute $(p,q)=(30, 20)$ , $d=11$ to the above formula:

Because there are $11$ lattice points on line $y = \frac{2}{3}x$ , if $m < \frac{2}{3}$ , then the number of lattice points on or below the line is less than $300$ . So $m = \frac{2}{3}$ is the lower bound.

Now I am going to calculate the upper bound. From $\frac{b}{a} < \frac{b+1}{a+1}$ ,

$\frac{2}{3} = \frac{18}{27} < \frac{19}{28}$

$\frac{2}{3} = \frac{20}{30} < \frac{21}{31} < \frac{19}{28}$

$[asy] /* Created by Brendanb4321, modified by isabelchen */ import graph; size(18cm); defaultpen(fontsize(9pt)); xaxis(0,31,Ticks(1.0)); yaxis(0,22,Ticks(1.0)); draw((0,0)--(30,20)); draw((0,0)--(30,30*19/28), dotted); draw((0,0)--(31,31*21/31), dotted); for (int i = 1; i<=31; ++i) { for (int j = 1; j<=2/3*i+1; ++j) { dot((i,j)); } } dot((28,19), red); dot((31,21), blue); label("$m=2/3$", (33,20)); label("$m=21/31$", (33,21)); label("$m=19/28$", (33,22)); [/asy]$

If $m = \frac{21}{31}$ , I will calculate by using the rectangle with blue vertex $(p,q) = (31, 21)$ , then subtract lattice points on line $x = 31$ , which is $21$ . There are 2 lattice points on the diagonal, $d=2$ .

, same as that of

If $m = \frac{19}{28}$ , I will calculate by using the rectangle with red vertex $(p,q) = (28, 19)$ , then add lattice points on line $x = 29$ and $x = 30$ , which is $19 + 20 = 39$ . There are 2 lattice points on the diagonal, $d=2$ .

,  more than that of

When $m$ increases, more lattice points falls below the line $y = mx$ . Any value larger than $\frac{19}{28}$ has more than $301$ lattice points on or below $y = \frac{19}{28} x$ . So the upper bound is $\frac{19}{28}$ .

$\frac{19}{28}-\frac{2}{3}=\frac{1}{84}$ , $\boxed{\textbf{(E)} ~85}$ .

~isabelchen

Solution 5

The lower bound of $m$ is $\frac23 = \frac{20}{30}$ . Inside the rectangle with vertices $(0,0)$ , $(30,0)$ , $(30,20)$ , $(0, 20)$ and diagonal $y = \frac23 x$ , there are $(30-1)(20-1) = 551$ lattice points inside, not including the edges. There are $9$ lattice points on diagonal $y = \frac23 x$ inside the rectangle, $551 + 9 = 560$ . Half of the $560$ lattice points are below diagonal $y = \frac23 x$ , $560 \cdot \frac12 = 280$ . There are $20$ lattice points on edge $x = 30$ , $280 + 20 = 300$ . Once $m < \frac23$ , the $9$ lattice points on diagonal $y = \frac23 x$ will be above the new diagonal, making the number of lattice points on and below the diagonal less than $300$ .

Now we are going to calculate the upper bound by the following formula:

The number of lattice points inside rectangle , , ,  and below diagonal  is , where  and  are relatively prime.
There are  lattice points inside the rectangle. Because  and  are relatively prime, the only lattice points on the diagonal are  and . By symmetry, half of the lattice points are below the diagonal.

$[asy] /* Created by isabelchen */ import graph; size(8cm); defaultpen(fontsize(9pt)); xaxis(0,8); yaxis(0,6); draw((0,0)--(7,5)); draw((7,0)--(7,5)); draw((0,5)--(7,5)); dot((0,0)); dot((1,0)); dot((2,0)); dot((3,0)); dot((4,0)); dot((5,0)); dot((6,0)); dot((7,0)); dot((8,0)); dot((0,1)); dot((1,1)); dot((2,1)); dot((3,1)); dot((4,1)); dot((5,1)); dot((6,1)); dot((7,1)); dot((8,1)); dot((0,2)); dot((1,2)); dot((2,2)); dot((3,2)); dot((4,2)); dot((5,2)); dot((6,2)); dot((7,2)); dot((8,2)); dot((0,3)); dot((1,3)); dot((2,3)); dot((3,3)); dot((4,3)); dot((5,3)); dot((6,3)); dot((7,3)); dot((8,3)); dot((0,4)); dot((1,4)); dot((2,4)); dot((3,4)); dot((4,4)); dot((5,4)); dot((6,4)); dot((7,4)); dot((8,4)); dot((0,5)); dot((1,5)); dot((2,5)); dot((3,5)); dot((4,5)); dot((5,5)); dot((6,5)); dot((7,5)); dot((8,5)); dot((0,6)); dot((1,6)); dot((2,6)); dot((3,6)); dot((4,6)); dot((5,6)); dot((6,6)); dot((7,6)); dot((8,6)); label("$(0,0)$", (0,0), SW); label("$(a, b)$", (7,5), NE); dot((7,0)); label("$a$", (7,0), S); dot((0,5)); label("$b$", (0,5), W); [/asy]$

$\frac{2}{3} = \frac{20}{30} < \frac{21}{31} < \frac{19}{28}$

When $a = 31$ , $b = 21$ , $\frac{(31-1)(21-1)}{2} = 300$ .

When $a = 28$ , $b = 19$ , $\frac{(28-1)(19-1)}{2} = 243$ . Below the line $y = \frac{19}{28} x$ , there are $19$ lattice points on line $x = 28$ , $19$ lattice points on line $x = 29$ , $20$ lattice points on line $x = 30$ , $243 + 19 + 19 + 20 = 301$ .

More lattice points fall below the line $y = mx$ as $m$ increases. There are more than $301$ lattice points on and below the line for any $m$ greater than $\frac{19}{28}$ . Therefore, the upper bound is $\frac{19}{28}$ .

$\frac{19}{28}-\frac{2}{3}=\frac{1}{84}$ , so $1+84=\boxed{\textbf{(E)} ~85}$ .

~isabelchen

Remark

$\lfloor \frac{b}{a} k \rfloor$ is the number of lattice points on line $x = k$ , below line $y = \frac{b}{a} x$ and above the $x$ axis, where $k$ is an integer and $0<k<a$ . Therefore, $\sum_{k=1}^{a-1} \lfloor \frac{b}{a} k \rfloor$ is the number of lattice points inside the rectangle $(0,0)$ , $(a,0)$ , $(a, b)$ , $(0, b)$ , below diagonal $y = \frac{b}{a} x$ . If $a$ and $b$ are relatively prime, $\sum_{k=1}^{a-1} \lfloor \frac{b}{a} k \rfloor = \frac{(a-1)(b-1)}{2}$ , as explained in solution 6. This problem is about finding the upper and lower bound of $\frac{b}{a}$ , given $\sum_{k=1}^{30} \lfloor \frac{b}{a} k \rfloor = 300$ . The same problem can have geometric representation as stated in the original problem, or algebraic representation as stated here.

~isabelchen

Video Solution

https://youtu.be/PC8fIZzICFg ~hippopotamus1

Video Solution by Interstigation (In-Depth, Straight-forward)

https://youtu.be/aAogEAOcL2Y

~Interstigation

@@ Line 37: / Line 37: @@
 ==Solution 2==
-I know that I want about <math>\frac{2}{3}</math> of the box of integer coordinates above my line. There are a total of 30 integer coordinates in the desired range for each axis which gives a total of <math>900</math> lattice points. I estimate that the slope, <math>m</math>, is <math>\frac{2}{3}</math>. Now, although there is probably an easier solution, I would try to count the number of points above the line to see if there are <math>600</math> points above the line. The line <math>y=\frac{2}{3}x</math> separates the area inside the box so that <math>\frac{2}{3}</math> of the area is above the line.
+I know that I want about <math>\frac{2}{3}</math> of the box of integer coordinates above my line. There are a total of <math>30</math> integer coordinates in the desired range for each axis which gives a total of <math>900</math> lattice points. I estimate that the slope, <math>m</math>, is <math>\frac{2}{3}</math>. Now, although there is probably an easier solution, I would try to count the number of points above the line to see if there are <math>600</math> points above the line. The line <math>y=\frac{2}{3}x</math> separates the area inside the box so that <math>\frac{2}{3}</math> of the area is above the line.
 I find that the number of coordinates with <math>x=1</math> above the line is <math>30</math>, and the number of coordinates with <math>x=2</math> above the line is 29. Every time the line <math>y=\frac{2}{3}x</math> hits a y-value with an integer coordinate, the number of points above the line decreases by one. I wrote out the sum of <math>30</math> terms in hopes of finding a pattern. I graphed the first couple positive integer x-coordinates, and found that the sum of the integers above the line is <math>30+29+28+28+27+26+26 \ldots+ 10</math>. The even integer repeats itself every third term in the sum. I found that the average of each of the terms is <math>20</math>, and there are <math>30</math> of them which means that exactly <math>600</math> above the line as desired. This give a lower bound because if the slope decreases a little bit, then the points that the line goes through will be above the line.

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Difference between revisions of "2021 AMC 12B Problems/Problem 25"

Revision as of 19:17, 16 July 2024

Contents

Problem

Solution 1

Solution 2

Diagram

Solution 3

Solution 4

Solution 5

Remark

Video Solution

Video Solution by Interstigation (In-Depth, Straight-forward)

See Also