https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=Theajl&feedformat=atomAoPS Wiki - User contributions [en]2024-03-28T15:45:57ZUser contributionsMediaWiki 1.31.1https://artofproblemsolving.com/wiki/index.php?title=2021_AMC_12B_Problems/Problem_25&diff=1466522021 AMC 12B Problems/Problem 252021-02-14T18:51:19Z<p>Theajl: /* Solution 2 */</p>
<hr />
<div>==Problem==<br />
<br />
Let <math>S</math> be the set of lattice points in the coordinate plane, both of whose coordinates are integers between <math>1</math> and <math>30,</math> inclusive. Exactly <math>300</math> points in <math>S</math> lie on or below a line with equation <math>y=mx.</math> The possible values of <math>m</math> lie in an interval of length <math>\frac ab,</math> where <math>a</math> and <math>b</math> are relatively prime positive integers. What is <math>a+b?</math><br />
<br />
<math>\textbf{(A)} ~31 \qquad \textbf{(B)} ~47 \qquad \textbf{(C)} ~62\qquad \textbf{(D)} ~72 \qquad \textbf{(E)} ~85</math><br />
<br />
<br />
==Video Solution , Very Easy==<br />
https://youtu.be/PC8fIZzICFg<br />
~hippopotamus1<br />
<br />
==Solution 1==<br />
First, we find a numerical representation for the number of lattice points in <math>S</math> that are under the line <math>y=mx. </math> For any value of <math>x,</math> the highest lattice point under <math>y=mx</math> is <math>\lfloor mx \rfloor. </math> Because every lattice point from <math>(x, 1)</math> to <math>(x, \lfloor mx \rfloor)</math> is under the line, the total number of lattice points under the line is <math>\sum_{x=1}^{30}(\lfloor mx \rfloor). </math><br />
<br />
Now, we proceed by finding lower and upper bounds for <math>m. </math> To find the lower bound, we start with an approximation. If <math>300</math> lattice points are below the line, then around <math>\frac{1}{3}</math> of the area formed by <math>S</math> is under the line. By using the formula for a triangle's area, we find that when <math>x=30, y \approx 20. </math> Solving for <math>m</math> assuming that <math>(30, 20)</math> is a point on the line, we get <math>m = \frac{2}{3}. </math> Plugging in <math>m</math> to <math>\sum_{x=1}^{30}(\lfloor mx \rfloor), </math> we get: <br />
<br />
<cmath>\sum_{x=1}^{30}(\lfloor \frac{2}{3}x \rfloor) = 0 + 1 + 2 + 2 + 3 + \cdots + 18 + 18 + 19 + 20</cmath><br />
<br />
We have a repeat every <math>3</math> values (every time <math>y=\frac{2}{3}x</math> goes through a lattice point). Thus, we can use arithmetic sequences to calculate the value above:<br />
<br />
<cmath>\sum_{x=1}^{30}(\lfloor \frac{2}{3}x \rfloor) = 0 + 1 + 2 + 2 + 3 + \cdots + 18 + 18 + 19 + 20</cmath><cmath>=\frac{20(21)}{2} + 2 + 4 + 6 + \cdots + 18 </cmath><cmath>=210 + \frac{20}{2}\cdot 9</cmath><cmath>=300</cmath><br />
<br />
This means that <math>\frac{2}{3}</math> is a possible value of <math>m. </math> Furthermore, it is the lower bound for <math>m. </math> This is because <math>y=\frac{2}{3}x</math> goes through many points (such as <math>(21, 14)</math>). If <math>m</math> was lower, <math>y=\frac{2}{3}x</math> would no longer go through some of these points, and there would be less than <math>300</math> lattice points under it. <br />
<br />
Now, we find an upper bound for <math>m. </math> Imagine increasing <math>m</math> slowly and rotati<br />
ng the line <math>y=mx, </math> starting from the lower bound of <math>m=\frac{2}{3}. </math>The upper bound for <math>m</math> occurs when <math>y=mx</math> 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). <br />
<br />
<br />
In other words, we are looking for the first <math>m > \frac{2}{3}</math> that is expressible as a ratio of positive integers <math>\frac{p}{q}</math> with <math>q \le 30. </math> For each <math>q=1,\dots,30</math>, the smallest multiple of <math>\frac{1}{q}</math> which exceeds <math>\frac{2}{3}</math> is <math>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}</math> respectively, and the smallest of these is <math>\frac{19}{28}. </math> <br />
Note: start listing the multiples of <math>\frac{1}{q}</math> from <math>\frac{21}{30}</math> and observe that they get further and further away from <math>\frac{2}{3}. </math><br />
Alternatively, see the method of finding upper bounds in solution 2. <br />
<br />
<br />
The lower bound is <math>\frac{2}{3}</math> and the upper bound is <math>\frac{19}{28}. </math> Their difference is <math>\frac{1}{84}, </math> so the answer is <math>1 + 84 = \boxed{85}. </math><br />
<br />
~JimY<br />
<br />
==Solution 2==<br />
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 900 lattice points. I estimate that the slope, m, 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 600 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 are is above the line. <br />
<br />
I find that the number of coordinates with <math>x=1</math> above the line is 30, 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 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 <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 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. <br />
<br />
To find the upper bound, notice that each point with an integer-valued x-coordinate is either <math>\frac{1}{3}</math> or <math>\frac{2}{3}</math> 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 <math>x=28, 29, 30</math> which the line <math>y=\frac{2}{3}x</math> intersects at <math>y= \frac{56}{3}, \frac{58}{3}, 20</math>. 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, <math>\frac{56}{3}</math>) is closer to the lattice point. The slope of the line which goes through both the origin and (28,19) is <math>y=\frac{19}{28}x</math>. This gives an upper bound of <math>m=\frac{19}{28}</math>. <br />
<br />
Taking the upper bound of m and subtracting the lower bound yields <math>\frac{19}{28}-\frac{2}{3}=\frac{1}{84}</math>. This is answer <math>1+84=</math> <math>\boxed{\textbf{(E)} ~85}</math>.<br />
<br />
~theAJL<br />
===Diagram===<br />
<asy><br />
/* Created by Brendanb4321 */<br />
import graph;<br />
size(16cm);<br />
defaultpen(fontsize(9pt));<br />
xaxis(0,30,Ticks(1.0));<br />
yaxis(0,25,Ticks(1.0));<br />
<br />
draw((0,0)--(30,20));<br />
draw((0,0)--(30,30/28*19), dotted);<br />
int c = 0;<br />
for (int i = 1; i<=30; ++i)<br />
{<br />
for (int j = 1; j<=2/3*i+1; ++j)<br />
{<br />
dot((i,j));<br />
}<br />
}<br />
dot((28,19), red);<br />
label("$m=2/3$", (32,20));<br />
label("$m=19/28$", (32.3,20.8));<br />
</asy><br />
<br />
==See also==<br />
{{AMC12 box|year=2021|ab=B|num-b=24|after=Last problem}}<br />
<br />
[[Category:Intermediate Geometry Problems]]<br />
{{AMC10 box|year=2021|ab=B|num-b=24|after=Last problem}}</div>Theajlhttps://artofproblemsolving.com/wiki/index.php?title=AMC_historical_results&diff=146354AMC historical results2021-02-13T03:50:10Z<p>Theajl: /* AMC 12B */</p>
<hr />
<div><!-- Post AMC statistics and lists of high scorers here so that the AMC page doesn't get cluttered. --><br />
This is the '''AMC historical results''' page. This page should include results for the [[AIME]] as well. For [[USAMO]] results, see [[USAMO historical results]].<br />
==2021==<br />
===AMC 10A===<br />
*Average score: <br />
*AIME floor:<br />
*Distinction:<br />
*Distinguished Honor Roll:<br />
<br />
===AMC 10B===<br />
*Average score: <br />
*AIME floor:<br />
*Distinction: <br />
*Distinguished Honor Roll:<br />
<br />
===AMC 12A===<br />
*Average score: <br />
*AIME floor: <br />
*Distinction: <br />
*Distinguished Honor Roll:<br />
<br />
===AMC 12B===<br />
*Average score: <br />
*AIME floor:<br />
*Distinction: <br />
*Distinguished Honor Roll:<br />
<br />
===AIME I===<br />
*Average score: <br />
*Median score: <br />
*USAMO cutoff: <br />
*USAJMO cutoff:<br />
===AIME II===<br />
*Average score: <br />
*Median score: <br />
*USAMO cutoff: <br />
*USAJMO cutoff:<br />
===AMC 8===<br />
*Average score:<br />
*Honor Roll: <br />
*DHR:<br />
<br />
==2020==<br />
===AMC 10A===<br />
*Average score: 64.29<br />
*AIME floor: 103.5<br />
*Distinction: 105<br />
*Distinguished Honor Roll: 124.5<br />
<br />
===AMC 10B===<br />
*Average score: 61.22<br />
*AIME floor: 102<br />
*Distinction: 103.5 <br />
*Distinguished Honor Roll: 120<br />
<br />
===AMC 12A===<br />
*Average score: 61.42<br />
*AIME floor: 87<br />
*Distinction: 100.5<br />
*Distinguished Honor Roll: 123<br />
<br />
===AMC 12B===<br />
*Average score: 60.47<br />
*AIME floor: 87<br />
*Distinction: 97.5<br />
*Distinguished Honor Roll: 120<br />
<br />
===AIME I===<br />
*Average score: 5.70<br />
*Median score: 6<br />
*USOMO cutoff: 233.5 (AMC 12A), 235 (AMC 12B)<br />
*USOJMO cutoff: 229.5 (AMC 10A), 230 (AMC 10B)<br />
<br />
===AIME II===<br />
Due to COVID-19, the 2020 AIME II was administered online and referred to as the AOIME.<br />
*Average score: 6.1259<br />
*Median score: 6<br />
*USOMO cutoff: 234 (AMC 12A), 234.5 (AMC 12B)<br />
*USOJMO cutoff: 233.5 (AMC 10A), 229.5 (AMC 10B)<br />
<br />
===AMC 8===<br />
*Average score: 10.01<br />
*Honor Roll: 18<br />
*DHR: 21<br />
<br />
==2019==<br />
===AMC 10A===<br />
*Average score: 51.69<br />
*Honor roll: 96<br />
*AIME floor: 103.5<br />
*DHR: 123<br />
<br />
===AMC 10B===<br />
*Average score: 58.47<br />
*Honor roll: 102<br />
*AIME floor: 108<br />
*Distinguished Honor Roll: 121.5<br />
<br />
===AMC 12A===<br />
*Average score: 49.22<br />
*AIME floor: 84<br />
*DHR: 121.5<br />
<br />
===AMC 12B===<br />
*Average score: 56.74<br />
*AIME floor: 94.5 <br />
*DHR: 123<br />
<br />
===AIME I===<br />
*Average score: 5.88<br />
*Median score: 6<br />
*USAMO cutoff: 220 (AMC 12A), 230.5 (AMC 12B)<br />
*USAJMO cutoff: 209.5 (AMC 10A), 216 (AMC 10B)<br />
<br />
===AIME II===<br />
*Average score: 6.47<br />
*Median score: 6<br />
*USAMO cutoff: 230.5 (AMC 12A), 236 (AMC 12B)<br />
*USAJMO cutoff: 216.5 (AMC 10A), 220.5 (AMC 10B)<br />
<br />
===AMC 8===<br />
*Average score: 9.43<br />
*Honor roll: 19<br />
*DHR: 23<br />
<br />
==2018==<br />
===AMC 10A===<br />
*Average score: 53.84<br />
*Honor roll: 100.5<br />
*AIME floor: 111<br />
*DHR: 127.5<br />
<br />
===AMC 10B===<br />
*Average score: 57.81<br />
*Honor roll: 97.5<br />
*AIME floor: 108<br />
*DHR: 123<br />
<br />
===AMC 12A===<br />
*Average score: 56.36<br />
*AIME floor: 93<br />
*DHR: 120<br />
<br />
===AMC 12B===<br />
*Average score: 57.85<br />
*AIME floor: 99<br />
*DHR: 126<br />
<br />
===AIME I===<br />
*Average score: 5.09<br />
*Median score: 5<br />
*USAMO cutoff: 215 (AMC 12A), 235 (AMC 12B)<br />
*USAJMO cutoff: 222 (AMC 10A), 212 (AMC 10B)<br />
<br />
===AIME II===<br />
*Average score: 5.48<br />
*Median score: 5<br />
*USAMO cutoff: 216 (AMC 12A), 230.5 (AMC 12B)<br />
*USAJMO cutoff: 222 (AMC 10A), 212 (AMC 10B)<br />
<br />
===AMC 8===<br />
*Average score: 8.51<br />
*Honor roll: 15<br />
*DHR: 19<br />
<br />
==2017==<br />
===AMC 10A===<br />
*Average score: 59.33<br />
*AIME floor: 112.5<br />
*DHR: 127.5<br />
<br />
===AMC 10B===<br />
*Average score: 66.56<br />
*AIME floor: 120<br />
*DHR: 136.5<br />
<br />
===AMC 12A===<br />
*Average score: 60.32<br />
*AIME floor: 96<br />
*DHR: 115.5<br />
<br />
===AMC 12B===<br />
*Average score: 58.35<br />
*AIME floor: 100<br />
*DHR: 129<br />
<br />
===AIME I===<br />
*Average score: 5.69<br />
*Median score: 5<br />
*USAMO cutoff: 225 (AMC 12A), 235 (AMC 12B)<br />
*USAJMO cutoff: 224.5 (AMC 10A), 233 (AMC 10B)<br />
<br />
===AIME II===<br />
*Average score: 5.64<br />
*Median score: 5<br />
*USAMO cutoff: 221 (AMC 12A), 230.5 (AMC 12B)<br />
*USAJMO cutoff: 219 (AMC 10A), 225 (AMC 10B)<br />
<br />
===AMC 8===<br />
*Average score: 8.96<br />
*Honor roll: 17<br />
*DHR: 20<br />
<br />
==2016==<br />
===AMC 10A===<br />
*Average score: 65.31<br />
*AIME floor: 110<br />
*DHR: 120<br />
<br />
===AMC 10B===<br />
*Average score: 65.40<br />
*AIME floor: 110<br />
*DHR: 124.5<br />
<br />
===AMC 12A===<br />
*Average score: 59.06<br />
*AIME floor: 93<br />
*DHR: 111<br />
<br />
===AMC 12B===<br />
*Average score: 67.96<br />
*AIME floor: 100.5<br />
*DHR: 127.5<br />
<br />
===AIME I===<br />
*Average score: 5.83<br />
*Median score: 6<br />
*USAMO cutoff: 220<br />
*USAJMO cutoff: 210.5<br />
<br />
===AIME II===<br />
*Average score: 4.43<br />
*Median score: 4<br />
*USAMO cutoff: 205<br />
*USAJMO cutoff: 200<br />
<br />
===AMC 8===<br />
*Average score: 9.36<br />
*Honor roll: 18<br />
*DHR: 22<br />
<br />
==2015==<br />
===AMC 10A===<br />
*Average score: 73.39<br />
*AIME floor: 106.5<br />
*DHR: 115.5<br />
<br />
===AMC 10B===<br />
*Average score: 76.10<br />
*AIME floor: 120<br />
*DHR: 132<br />
<br />
===AMC 12A===<br />
*Average score: 69.90<br />
*AIME floor: 99<br />
*DHR: 117<br />
<br />
===AMC 12B===<br />
*Average score: 66.92<br />
*AIME floor: 100.5<br />
*DHR: 126<br />
<br />
===AIME I===<br />
*Average score: 5.29<br />
*Median score: 5<br />
*USAMO cutoff: 219.0<br />
*USAJMO cutoff: 213.0<br />
<br />
===AIME II===<br />
*Average score: 6.63<br />
*Median score: 6<br />
*USAMO cutoff: 229.0<br />
*USAJMO cutoff: 223.5<br />
<br />
===AMC 8===<br />
*Average score: 8.55<br />
*Honor roll: 16<br />
*DHR: 21<br />
<br />
==2014==<br />
===AMC 10A===<br />
*Average score: 63.83<br />
*AIME floor: 120<br />
*DHR: 132<br />
<br />
===AMC 10B===<br />
*Average score: 71.44<br />
*AIME floor: 120<br />
*DHR: 132<br />
<br />
===AMC 12A===<br />
*Average score: 64.01<br />
*AIME floor: 93<br />
*DHR: 109.5<br />
<br />
===AMC 12B===<br />
*Average score: 68.11<br />
*AIME floor: 100.5<br />
*DHR: 121.5<br />
<br />
===AIME I===<br />
*Average score: 4.88<br />
*Median score: 5<br />
*USAMO cutoff: 211.5<br />
*USAJMO cutoff: 211<br />
<br />
===AIME II===<br />
*Average score: 5.49<br />
*Median score: 5<br />
*USAMO cutoff: 211.5<br />
*USAJMO cutoff: 211<br />
<br />
===AMC 8===<br />
*Average score: 11.43<br />
*Honor roll: 19<br />
*DHR: 23<br />
<br />
==2013==<br />
===AMC 10A===<br />
*Average score: 72.50<br />
*AIME floor: 108<br />
*DHR: 117<br />
<br />
===AMC 10B===<br />
*Average score: 72.62<br />
*AIME floor: 120<br />
*DHR: 129<br />
<br />
===AMC 12A===<br />
*Average score: 65.06<br />
*AIME floor: 88.5<br />
*DHR: 106.5<br />
<br />
===AMC 12B===<br />
*Average score: 64.21<br />
*AIME floor: 93<br />
*DHR: 108<br />
<br />
===AIME I===<br />
*Average score: 4.69<br />
*Median score: 4<br />
*USAMO cutoff: 209<br />
*USAJMO cutoff: 210.5<br />
<br />
===AIME II===<br />
*Average score: 6.56<br />
*Median score: 6<br />
*USAMO cutoff: 209<br />
*USAJMO cutoff: 210.5<br />
<br />
===AMC 8===<br />
*Average score: 10.69<br />
*Honor roll: 18<br />
*DHR: 22<br />
<br />
==2012==<br />
===AMC 10A===<br />
*Average score: 72.51<br />
*AIME floor: 115.5<br />
*DHR: 121.5<br />
<br />
===AMC 10B===<br />
*Average score: 76.59<br />
*AIME floor: 120<br />
*DHR: 133.5<br />
<br />
===AMC 12A===<br />
*Average score: 64.62<br />
*AIME floor: 94.5<br />
*DHR: 109.5<br />
<br />
===AMC 12B===<br />
*Average score: 70.08<br />
*AIME floor: 99<br />
*DHR: 114<br />
<br />
===AIME I===<br />
*Average score: 5.13<br />
*Median score: 5<br />
*USAMO cutoff: 204.5<br />
*USAJMO cutoff: 204<br />
<br />
===AIME II===<br />
*Average score: 4.94<br />
*Median score: 5<br />
*USAMO cutoff: 204.5<br />
*USAJMO cutoff: 204<br />
<br />
===AMC 8===<br />
*Average score: 10.67<br />
*Honor roll: 18<br />
*DHR: 22<br />
<br />
==2011==<br />
===AMC 10A===<br />
*Average score: 64.24<br />
*AIME floor: 117<br />
*DHR: 129<br />
<br />
===AMC 10B===<br />
*Average score: 71.78<br />
*AIME floor: 117<br />
*DHR: 133.5<br />
<br />
===AMC 12A===<br />
*Average score: 65.38<br />
*AIME floor: 93<br />
*DHR: 112.5<br />
<br />
===AMC 12B===<br />
*Average score: 64.71<br />
*AIME floor: 97.5<br />
*DHR: 121.5<br />
<br />
===AIME I===<br />
*Average score: 2.23<br />
*Median score: 2<br />
*USAMO cutoff: 188<br />
*USAJMO cutoff: 179<br />
<br />
===AIME II===<br />
*Average score: 5.47<br />
*Median score: 5<br />
*USAMO cutoff: 215.5<br />
*USAJMO cutoff: 196.5<br />
<br />
===AMC 8===<br />
*Average score: 10.75<br />
*Honor roll: 17<br />
*DHR: 22<br />
<br />
==2010==<br />
===AMC 10A===<br />
*Average score: 68.11<br />
*AIME floor: 115.5<br />
<br />
===AMC 10B===<br />
*Average score: 68.57<br />
*AIME floor: 118.5<br />
<br />
===AMC 12A===<br />
*Average score: 61.02<br />
*AIME floor: 88.5<br />
*DHR cutoff: 108<br />
<br />
===AMC 12B===<br />
*Average score: 59.58<br />
*AIME floor: 88.5<br />
*DHR cutoff: 109.5<br />
<br />
===AIME I===<br />
*Average score: 5.90<br />
*Median score: 6<br />
*USAMO cutoff: 208.5 (204.5 for non juniors and seniors)<br />
*USAJMO cutoff: 188.5<br />
<br />
===AIME II===<br />
*Average score: 3.39<br />
*Median score: 3<br />
*USAMO cutoff: 208.5 (204.5 for non juniors and seniors)<br />
*USAJMO cutoff: 188.5<br />
<br />
===AMC 8===<br />
*Average score: 9.59<br />
*Honor roll: 17<br />
*DHR: 22<br />
<br />
==2009==<br />
===AMC 10A===<br />
*Average score: 67.41<br />
*AIME floor: 117<br />
<br />
===AMC 10B===<br />
*Average score: 74.73<br />
*AIME floor: 120<br />
<br />
===AMC 12A===<br />
*Average score: 66.37<br />
*AIME floor: 97.5<br />
<br />
===AMC 12B===<br />
*Average score: 71.88<br />
*AIME floor: 100 (Top 5% (1.00))<br />
<br />
===AIME I===<br />
*Average score: 4.17<br />
*Median score: 4<br />
*USAMO floor: <br />
<br />
===AIME II===<br />
*Average score: 3.27<br />
*Median score: 3<br />
*USAMO floor:<br />
<br />
===AMC 8===<br />
*Average score: 10.28<br />
*Honor roll: 17<br />
*DHR: 20<br />
<br />
==2008==<br />
===AMC 10A===<br />
*Average score: 60.25<br />
*AIME floor: 117<br />
<br />
===AMC 10B===<br />
*Average score: <br />
*AIME floor: 120<br />
<br />
===AMC 12A===<br />
*Average score: 65.6<br />
*AIME floor: 97.5<br />
<br />
===AMC 12B===<br />
*Average score: 68.9<br />
*AIME floor: 97.5<br />
<br />
===AIME I===<br />
*Average score: 4.77<br />
*Median score: 4<br />
*USAMO floor: <br />
<br />
===AIME II===<br />
*Average score: 5.27<br />
*Median score: 5<br />
*USAMO floor:<br />
<br />
===AMC 8===<br />
*Average score: 11.45<br />
*Honor roll: 19<br />
*DHR: 22<br />
<br />
==2007==<br />
<br />
===AMC 10A===<br />
*Average score: 67.9<br />
*AIME floor: 117<br />
<br />
===AMC 10B=== <br />
*Average score: 61.5<br />
*AIME floor: 115.5<br />
<br />
===AMC 12A===<br />
*Average score: 66.8<br />
*AIME floor: 97.5<br />
<br />
===AMC 12B===<br />
*Average score: 73.1<br />
*AIME floor: 100<br />
<br />
===AIME I===<br />
*Average score: 5<br />
*Median score: 3<br />
*USAMO floor: 6<br />
<br />
===AIME II===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
===AMC 8===<br />
*Average score: 9.87<br />
*Honor roll: 17<br />
*DHR: 21<br />
<br />
==2006==<br />
===AMC 10A===<br />
*Average score: 79.0<br />
*AIME floor: 120<br />
<br />
===AMC 10B===<br />
*Average score: 68.5<br />
*AIME floor: 120<br />
<br />
===AMC 12A===<br />
*Average score: 85.7<br />
*AIME floor: 100<br />
<br />
===AMC 12B===<br />
*Average score: 85.5<br />
*AIME floor: 100<br />
<br />
===AIME I===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
===AIME II===<br />
*Average score: <br />
*Median score:<br />
*USAMO floor:<br />
<br />
===AMC 8===<br />
*Honor roll: 17<br />
*DHR: 21<br />
<br />
==2005==<br />
===AMC 10A===<br />
*Average score: 74.0<br />
*AIME floor: 120<br />
<br />
===AMC 10B===<br />
*Average score: 79.0<br />
*AIME floor: 120<br />
<br />
===AMC 12A===<br />
*Average score: 78.7<br />
*AIME floor: 100<br />
<br />
===AMC 12B===<br />
*Average score: 83.4<br />
*AIME floor: 100<br />
<br />
===AIME I===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
===AIME II===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
===AMC 8===<br />
*Honor roll: 16<br />
*DHR: 20<br />
<br />
==2004==<br />
===AMC 10A===<br />
*Average score: 69.1<br />
*AIME floor: 110<br />
<br />
===AMC 10B===<br />
*Average score: 80.4<br />
*AIME floor: 115<br />
<br />
===AMC 12A===<br />
*Average score: 73.9<br />
*AIME floor: 100<br />
<br />
===AMC 12B===<br />
*Average score: 84.5<br />
*AIME floor: 100<br />
<br />
===AIME I===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
===AIME II===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
===AMC 8===<br />
*Honor roll: 17<br />
*DHR: 21<br />
<br />
==2003==<br />
===AMC 10A===<br />
*Average score: 74.4<br />
*AIME floor: 119<br />
<br />
===AMC 10B===<br />
*Average score: 79.6<br />
*AIME floor: 121<br />
<br />
===AMC 12A===<br />
*Average score: 77.8<br />
*AIME floor: 100<br />
<br />
===AMC 12B===<br />
*Average score: 76.6<br />
*AIME floor: 100<br />
<br />
===AIME I===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
===AIME II===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
===AMC 8===<br />
*Honor roll: 18<br />
*DHR: 22<br />
<br />
==2002==<br />
===AMC 10A===<br />
*Average score: 68.5<br />
*AIME floor: 115<br />
<br />
===AMC 10B===<br />
*Average score: 74.9<br />
*AIME floor: 118<br />
<br />
===AMC 12A===<br />
*Average score: 72.7<br />
*AIME floor: 100<br />
<br />
===AMC 12B===<br />
*Average score: 80.8<br />
*AIME floor: 100<br />
<br />
===AIME I===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
===AIME II===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
==2001==<br />
===AMC 10===<br />
*Average score: 67.8<br />
*AIME floor: 116<br />
<br />
===AMC 12===<br />
*Average score: 56.6<br />
*AIME floor: 84<br />
<br />
===AIME I===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
===AIME II===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
==2000==<br />
===AMC 10===<br />
*Average score: <math>64.2</math><br />
*AIME floor: <math>110</math><br />
<br />
===AMC 12===<br />
*Average score: <math>64.9</math><br />
*AIME floor: <math>92</math><br />
<br />
===AIME I===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
===AIME II===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
==1999==<br />
===AHSME===<br />
*Average score: <math>68.8</math><br />
*AIME floor:<br />
<br />
===AIME===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:</div>Theajlhttps://artofproblemsolving.com/wiki/index.php?title=AMC_historical_results&diff=146353AMC historical results2021-02-13T03:49:50Z<p>Theajl: /* AMC 12A */</p>
<hr />
<div><!-- Post AMC statistics and lists of high scorers here so that the AMC page doesn't get cluttered. --><br />
This is the '''AMC historical results''' page. This page should include results for the [[AIME]] as well. For [[USAMO]] results, see [[USAMO historical results]].<br />
==2021==<br />
===AMC 10A===<br />
*Average score: <br />
*AIME floor:<br />
*Distinction:<br />
*Distinguished Honor Roll:<br />
<br />
===AMC 10B===<br />
*Average score: <br />
*AIME floor:<br />
*Distinction: <br />
*Distinguished Honor Roll:<br />
<br />
===AMC 12A===<br />
*Average score: <br />
*AIME floor: <br />
*Distinction: <br />
*Distinguished Honor Roll:<br />
<br />
===AMC 12B===<br />
*Average score: 33.3333333<br />
*AIME floor: 69 (Expanded due to cheating concerns.)<br />
*Distinction: 4<br />
*Distinguished Honor Roll: 20<br />
===AIME I===<br />
*Average score: <br />
*Median score: <br />
*USAMO cutoff: <br />
*USAJMO cutoff:<br />
===AIME II===<br />
*Average score: <br />
*Median score: <br />
*USAMO cutoff: <br />
*USAJMO cutoff:<br />
===AMC 8===<br />
*Average score:<br />
*Honor Roll: <br />
*DHR:<br />
<br />
==2020==<br />
===AMC 10A===<br />
*Average score: 64.29<br />
*AIME floor: 103.5<br />
*Distinction: 105<br />
*Distinguished Honor Roll: 124.5<br />
<br />
===AMC 10B===<br />
*Average score: 61.22<br />
*AIME floor: 102<br />
*Distinction: 103.5 <br />
*Distinguished Honor Roll: 120<br />
<br />
===AMC 12A===<br />
*Average score: 61.42<br />
*AIME floor: 87<br />
*Distinction: 100.5<br />
*Distinguished Honor Roll: 123<br />
<br />
===AMC 12B===<br />
*Average score: 60.47<br />
*AIME floor: 87<br />
*Distinction: 97.5<br />
*Distinguished Honor Roll: 120<br />
<br />
===AIME I===<br />
*Average score: 5.70<br />
*Median score: 6<br />
*USOMO cutoff: 233.5 (AMC 12A), 235 (AMC 12B)<br />
*USOJMO cutoff: 229.5 (AMC 10A), 230 (AMC 10B)<br />
<br />
===AIME II===<br />
Due to COVID-19, the 2020 AIME II was administered online and referred to as the AOIME.<br />
*Average score: 6.1259<br />
*Median score: 6<br />
*USOMO cutoff: 234 (AMC 12A), 234.5 (AMC 12B)<br />
*USOJMO cutoff: 233.5 (AMC 10A), 229.5 (AMC 10B)<br />
<br />
===AMC 8===<br />
*Average score: 10.01<br />
*Honor Roll: 18<br />
*DHR: 21<br />
<br />
==2019==<br />
===AMC 10A===<br />
*Average score: 51.69<br />
*Honor roll: 96<br />
*AIME floor: 103.5<br />
*DHR: 123<br />
<br />
===AMC 10B===<br />
*Average score: 58.47<br />
*Honor roll: 102<br />
*AIME floor: 108<br />
*Distinguished Honor Roll: 121.5<br />
<br />
===AMC 12A===<br />
*Average score: 49.22<br />
*AIME floor: 84<br />
*DHR: 121.5<br />
<br />
===AMC 12B===<br />
*Average score: 56.74<br />
*AIME floor: 94.5 <br />
*DHR: 123<br />
<br />
===AIME I===<br />
*Average score: 5.88<br />
*Median score: 6<br />
*USAMO cutoff: 220 (AMC 12A), 230.5 (AMC 12B)<br />
*USAJMO cutoff: 209.5 (AMC 10A), 216 (AMC 10B)<br />
<br />
===AIME II===<br />
*Average score: 6.47<br />
*Median score: 6<br />
*USAMO cutoff: 230.5 (AMC 12A), 236 (AMC 12B)<br />
*USAJMO cutoff: 216.5 (AMC 10A), 220.5 (AMC 10B)<br />
<br />
===AMC 8===<br />
*Average score: 9.43<br />
*Honor roll: 19<br />
*DHR: 23<br />
<br />
==2018==<br />
===AMC 10A===<br />
*Average score: 53.84<br />
*Honor roll: 100.5<br />
*AIME floor: 111<br />
*DHR: 127.5<br />
<br />
===AMC 10B===<br />
*Average score: 57.81<br />
*Honor roll: 97.5<br />
*AIME floor: 108<br />
*DHR: 123<br />
<br />
===AMC 12A===<br />
*Average score: 56.36<br />
*AIME floor: 93<br />
*DHR: 120<br />
<br />
===AMC 12B===<br />
*Average score: 57.85<br />
*AIME floor: 99<br />
*DHR: 126<br />
<br />
===AIME I===<br />
*Average score: 5.09<br />
*Median score: 5<br />
*USAMO cutoff: 215 (AMC 12A), 235 (AMC 12B)<br />
*USAJMO cutoff: 222 (AMC 10A), 212 (AMC 10B)<br />
<br />
===AIME II===<br />
*Average score: 5.48<br />
*Median score: 5<br />
*USAMO cutoff: 216 (AMC 12A), 230.5 (AMC 12B)<br />
*USAJMO cutoff: 222 (AMC 10A), 212 (AMC 10B)<br />
<br />
===AMC 8===<br />
*Average score: 8.51<br />
*Honor roll: 15<br />
*DHR: 19<br />
<br />
==2017==<br />
===AMC 10A===<br />
*Average score: 59.33<br />
*AIME floor: 112.5<br />
*DHR: 127.5<br />
<br />
===AMC 10B===<br />
*Average score: 66.56<br />
*AIME floor: 120<br />
*DHR: 136.5<br />
<br />
===AMC 12A===<br />
*Average score: 60.32<br />
*AIME floor: 96<br />
*DHR: 115.5<br />
<br />
===AMC 12B===<br />
*Average score: 58.35<br />
*AIME floor: 100<br />
*DHR: 129<br />
<br />
===AIME I===<br />
*Average score: 5.69<br />
*Median score: 5<br />
*USAMO cutoff: 225 (AMC 12A), 235 (AMC 12B)<br />
*USAJMO cutoff: 224.5 (AMC 10A), 233 (AMC 10B)<br />
<br />
===AIME II===<br />
*Average score: 5.64<br />
*Median score: 5<br />
*USAMO cutoff: 221 (AMC 12A), 230.5 (AMC 12B)<br />
*USAJMO cutoff: 219 (AMC 10A), 225 (AMC 10B)<br />
<br />
===AMC 8===<br />
*Average score: 8.96<br />
*Honor roll: 17<br />
*DHR: 20<br />
<br />
==2016==<br />
===AMC 10A===<br />
*Average score: 65.31<br />
*AIME floor: 110<br />
*DHR: 120<br />
<br />
===AMC 10B===<br />
*Average score: 65.40<br />
*AIME floor: 110<br />
*DHR: 124.5<br />
<br />
===AMC 12A===<br />
*Average score: 59.06<br />
*AIME floor: 93<br />
*DHR: 111<br />
<br />
===AMC 12B===<br />
*Average score: 67.96<br />
*AIME floor: 100.5<br />
*DHR: 127.5<br />
<br />
===AIME I===<br />
*Average score: 5.83<br />
*Median score: 6<br />
*USAMO cutoff: 220<br />
*USAJMO cutoff: 210.5<br />
<br />
===AIME II===<br />
*Average score: 4.43<br />
*Median score: 4<br />
*USAMO cutoff: 205<br />
*USAJMO cutoff: 200<br />
<br />
===AMC 8===<br />
*Average score: 9.36<br />
*Honor roll: 18<br />
*DHR: 22<br />
<br />
==2015==<br />
===AMC 10A===<br />
*Average score: 73.39<br />
*AIME floor: 106.5<br />
*DHR: 115.5<br />
<br />
===AMC 10B===<br />
*Average score: 76.10<br />
*AIME floor: 120<br />
*DHR: 132<br />
<br />
===AMC 12A===<br />
*Average score: 69.90<br />
*AIME floor: 99<br />
*DHR: 117<br />
<br />
===AMC 12B===<br />
*Average score: 66.92<br />
*AIME floor: 100.5<br />
*DHR: 126<br />
<br />
===AIME I===<br />
*Average score: 5.29<br />
*Median score: 5<br />
*USAMO cutoff: 219.0<br />
*USAJMO cutoff: 213.0<br />
<br />
===AIME II===<br />
*Average score: 6.63<br />
*Median score: 6<br />
*USAMO cutoff: 229.0<br />
*USAJMO cutoff: 223.5<br />
<br />
===AMC 8===<br />
*Average score: 8.55<br />
*Honor roll: 16<br />
*DHR: 21<br />
<br />
==2014==<br />
===AMC 10A===<br />
*Average score: 63.83<br />
*AIME floor: 120<br />
*DHR: 132<br />
<br />
===AMC 10B===<br />
*Average score: 71.44<br />
*AIME floor: 120<br />
*DHR: 132<br />
<br />
===AMC 12A===<br />
*Average score: 64.01<br />
*AIME floor: 93<br />
*DHR: 109.5<br />
<br />
===AMC 12B===<br />
*Average score: 68.11<br />
*AIME floor: 100.5<br />
*DHR: 121.5<br />
<br />
===AIME I===<br />
*Average score: 4.88<br />
*Median score: 5<br />
*USAMO cutoff: 211.5<br />
*USAJMO cutoff: 211<br />
<br />
===AIME II===<br />
*Average score: 5.49<br />
*Median score: 5<br />
*USAMO cutoff: 211.5<br />
*USAJMO cutoff: 211<br />
<br />
===AMC 8===<br />
*Average score: 11.43<br />
*Honor roll: 19<br />
*DHR: 23<br />
<br />
==2013==<br />
===AMC 10A===<br />
*Average score: 72.50<br />
*AIME floor: 108<br />
*DHR: 117<br />
<br />
===AMC 10B===<br />
*Average score: 72.62<br />
*AIME floor: 120<br />
*DHR: 129<br />
<br />
===AMC 12A===<br />
*Average score: 65.06<br />
*AIME floor: 88.5<br />
*DHR: 106.5<br />
<br />
===AMC 12B===<br />
*Average score: 64.21<br />
*AIME floor: 93<br />
*DHR: 108<br />
<br />
===AIME I===<br />
*Average score: 4.69<br />
*Median score: 4<br />
*USAMO cutoff: 209<br />
*USAJMO cutoff: 210.5<br />
<br />
===AIME II===<br />
*Average score: 6.56<br />
*Median score: 6<br />
*USAMO cutoff: 209<br />
*USAJMO cutoff: 210.5<br />
<br />
===AMC 8===<br />
*Average score: 10.69<br />
*Honor roll: 18<br />
*DHR: 22<br />
<br />
==2012==<br />
===AMC 10A===<br />
*Average score: 72.51<br />
*AIME floor: 115.5<br />
*DHR: 121.5<br />
<br />
===AMC 10B===<br />
*Average score: 76.59<br />
*AIME floor: 120<br />
*DHR: 133.5<br />
<br />
===AMC 12A===<br />
*Average score: 64.62<br />
*AIME floor: 94.5<br />
*DHR: 109.5<br />
<br />
===AMC 12B===<br />
*Average score: 70.08<br />
*AIME floor: 99<br />
*DHR: 114<br />
<br />
===AIME I===<br />
*Average score: 5.13<br />
*Median score: 5<br />
*USAMO cutoff: 204.5<br />
*USAJMO cutoff: 204<br />
<br />
===AIME II===<br />
*Average score: 4.94<br />
*Median score: 5<br />
*USAMO cutoff: 204.5<br />
*USAJMO cutoff: 204<br />
<br />
===AMC 8===<br />
*Average score: 10.67<br />
*Honor roll: 18<br />
*DHR: 22<br />
<br />
==2011==<br />
===AMC 10A===<br />
*Average score: 64.24<br />
*AIME floor: 117<br />
*DHR: 129<br />
<br />
===AMC 10B===<br />
*Average score: 71.78<br />
*AIME floor: 117<br />
*DHR: 133.5<br />
<br />
===AMC 12A===<br />
*Average score: 65.38<br />
*AIME floor: 93<br />
*DHR: 112.5<br />
<br />
===AMC 12B===<br />
*Average score: 64.71<br />
*AIME floor: 97.5<br />
*DHR: 121.5<br />
<br />
===AIME I===<br />
*Average score: 2.23<br />
*Median score: 2<br />
*USAMO cutoff: 188<br />
*USAJMO cutoff: 179<br />
<br />
===AIME II===<br />
*Average score: 5.47<br />
*Median score: 5<br />
*USAMO cutoff: 215.5<br />
*USAJMO cutoff: 196.5<br />
<br />
===AMC 8===<br />
*Average score: 10.75<br />
*Honor roll: 17<br />
*DHR: 22<br />
<br />
==2010==<br />
===AMC 10A===<br />
*Average score: 68.11<br />
*AIME floor: 115.5<br />
<br />
===AMC 10B===<br />
*Average score: 68.57<br />
*AIME floor: 118.5<br />
<br />
===AMC 12A===<br />
*Average score: 61.02<br />
*AIME floor: 88.5<br />
*DHR cutoff: 108<br />
<br />
===AMC 12B===<br />
*Average score: 59.58<br />
*AIME floor: 88.5<br />
*DHR cutoff: 109.5<br />
<br />
===AIME I===<br />
*Average score: 5.90<br />
*Median score: 6<br />
*USAMO cutoff: 208.5 (204.5 for non juniors and seniors)<br />
*USAJMO cutoff: 188.5<br />
<br />
===AIME II===<br />
*Average score: 3.39<br />
*Median score: 3<br />
*USAMO cutoff: 208.5 (204.5 for non juniors and seniors)<br />
*USAJMO cutoff: 188.5<br />
<br />
===AMC 8===<br />
*Average score: 9.59<br />
*Honor roll: 17<br />
*DHR: 22<br />
<br />
==2009==<br />
===AMC 10A===<br />
*Average score: 67.41<br />
*AIME floor: 117<br />
<br />
===AMC 10B===<br />
*Average score: 74.73<br />
*AIME floor: 120<br />
<br />
===AMC 12A===<br />
*Average score: 66.37<br />
*AIME floor: 97.5<br />
<br />
===AMC 12B===<br />
*Average score: 71.88<br />
*AIME floor: 100 (Top 5% (1.00))<br />
<br />
===AIME I===<br />
*Average score: 4.17<br />
*Median score: 4<br />
*USAMO floor: <br />
<br />
===AIME II===<br />
*Average score: 3.27<br />
*Median score: 3<br />
*USAMO floor:<br />
<br />
===AMC 8===<br />
*Average score: 10.28<br />
*Honor roll: 17<br />
*DHR: 20<br />
<br />
==2008==<br />
===AMC 10A===<br />
*Average score: 60.25<br />
*AIME floor: 117<br />
<br />
===AMC 10B===<br />
*Average score: <br />
*AIME floor: 120<br />
<br />
===AMC 12A===<br />
*Average score: 65.6<br />
*AIME floor: 97.5<br />
<br />
===AMC 12B===<br />
*Average score: 68.9<br />
*AIME floor: 97.5<br />
<br />
===AIME I===<br />
*Average score: 4.77<br />
*Median score: 4<br />
*USAMO floor: <br />
<br />
===AIME II===<br />
*Average score: 5.27<br />
*Median score: 5<br />
*USAMO floor:<br />
<br />
===AMC 8===<br />
*Average score: 11.45<br />
*Honor roll: 19<br />
*DHR: 22<br />
<br />
==2007==<br />
<br />
===AMC 10A===<br />
*Average score: 67.9<br />
*AIME floor: 117<br />
<br />
===AMC 10B=== <br />
*Average score: 61.5<br />
*AIME floor: 115.5<br />
<br />
===AMC 12A===<br />
*Average score: 66.8<br />
*AIME floor: 97.5<br />
<br />
===AMC 12B===<br />
*Average score: 73.1<br />
*AIME floor: 100<br />
<br />
===AIME I===<br />
*Average score: 5<br />
*Median score: 3<br />
*USAMO floor: 6<br />
<br />
===AIME II===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
===AMC 8===<br />
*Average score: 9.87<br />
*Honor roll: 17<br />
*DHR: 21<br />
<br />
==2006==<br />
===AMC 10A===<br />
*Average score: 79.0<br />
*AIME floor: 120<br />
<br />
===AMC 10B===<br />
*Average score: 68.5<br />
*AIME floor: 120<br />
<br />
===AMC 12A===<br />
*Average score: 85.7<br />
*AIME floor: 100<br />
<br />
===AMC 12B===<br />
*Average score: 85.5<br />
*AIME floor: 100<br />
<br />
===AIME I===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
===AIME II===<br />
*Average score: <br />
*Median score:<br />
*USAMO floor:<br />
<br />
===AMC 8===<br />
*Honor roll: 17<br />
*DHR: 21<br />
<br />
==2005==<br />
===AMC 10A===<br />
*Average score: 74.0<br />
*AIME floor: 120<br />
<br />
===AMC 10B===<br />
*Average score: 79.0<br />
*AIME floor: 120<br />
<br />
===AMC 12A===<br />
*Average score: 78.7<br />
*AIME floor: 100<br />
<br />
===AMC 12B===<br />
*Average score: 83.4<br />
*AIME floor: 100<br />
<br />
===AIME I===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
===AIME II===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
===AMC 8===<br />
*Honor roll: 16<br />
*DHR: 20<br />
<br />
==2004==<br />
===AMC 10A===<br />
*Average score: 69.1<br />
*AIME floor: 110<br />
<br />
===AMC 10B===<br />
*Average score: 80.4<br />
*AIME floor: 115<br />
<br />
===AMC 12A===<br />
*Average score: 73.9<br />
*AIME floor: 100<br />
<br />
===AMC 12B===<br />
*Average score: 84.5<br />
*AIME floor: 100<br />
<br />
===AIME I===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
===AIME II===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
===AMC 8===<br />
*Honor roll: 17<br />
*DHR: 21<br />
<br />
==2003==<br />
===AMC 10A===<br />
*Average score: 74.4<br />
*AIME floor: 119<br />
<br />
===AMC 10B===<br />
*Average score: 79.6<br />
*AIME floor: 121<br />
<br />
===AMC 12A===<br />
*Average score: 77.8<br />
*AIME floor: 100<br />
<br />
===AMC 12B===<br />
*Average score: 76.6<br />
*AIME floor: 100<br />
<br />
===AIME I===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
===AIME II===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
===AMC 8===<br />
*Honor roll: 18<br />
*DHR: 22<br />
<br />
==2002==<br />
===AMC 10A===<br />
*Average score: 68.5<br />
*AIME floor: 115<br />
<br />
===AMC 10B===<br />
*Average score: 74.9<br />
*AIME floor: 118<br />
<br />
===AMC 12A===<br />
*Average score: 72.7<br />
*AIME floor: 100<br />
<br />
===AMC 12B===<br />
*Average score: 80.8<br />
*AIME floor: 100<br />
<br />
===AIME I===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
===AIME II===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
==2001==<br />
===AMC 10===<br />
*Average score: 67.8<br />
*AIME floor: 116<br />
<br />
===AMC 12===<br />
*Average score: 56.6<br />
*AIME floor: 84<br />
<br />
===AIME I===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
===AIME II===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
==2000==<br />
===AMC 10===<br />
*Average score: <math>64.2</math><br />
*AIME floor: <math>110</math><br />
<br />
===AMC 12===<br />
*Average score: <math>64.9</math><br />
*AIME floor: <math>92</math><br />
<br />
===AIME I===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
===AIME II===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
==1999==<br />
===AHSME===<br />
*Average score: <math>68.8</math><br />
*AIME floor:<br />
<br />
===AIME===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:</div>Theajlhttps://artofproblemsolving.com/wiki/index.php?title=AMC_historical_results&diff=146351AMC historical results2021-02-13T03:49:28Z<p>Theajl: /* AMC 10B */</p>
<hr />
<div><!-- Post AMC statistics and lists of high scorers here so that the AMC page doesn't get cluttered. --><br />
This is the '''AMC historical results''' page. This page should include results for the [[AIME]] as well. For [[USAMO]] results, see [[USAMO historical results]].<br />
==2021==<br />
===AMC 10A===<br />
*Average score: <br />
*AIME floor:<br />
*Distinction:<br />
*Distinguished Honor Roll:<br />
<br />
===AMC 10B===<br />
*Average score: <br />
*AIME floor:<br />
*Distinction: <br />
*Distinguished Honor Roll:<br />
<br />
===AMC 12A===<br />
*Average score: 36<br />
*AIME floor: 37.5 (Expanded due to cheating concerns.)<br />
*Distinction: 42<br />
*Distinguished Honor Roll: 43<br />
<br />
===AMC 12B===<br />
*Average score: 33.3333333<br />
*AIME floor: 69 (Expanded due to cheating concerns.)<br />
*Distinction: 4<br />
*Distinguished Honor Roll: 20<br />
===AIME I===<br />
*Average score: <br />
*Median score: <br />
*USAMO cutoff: <br />
*USAJMO cutoff:<br />
===AIME II===<br />
*Average score: <br />
*Median score: <br />
*USAMO cutoff: <br />
*USAJMO cutoff:<br />
===AMC 8===<br />
*Average score:<br />
*Honor Roll: <br />
*DHR:<br />
<br />
==2020==<br />
===AMC 10A===<br />
*Average score: 64.29<br />
*AIME floor: 103.5<br />
*Distinction: 105<br />
*Distinguished Honor Roll: 124.5<br />
<br />
===AMC 10B===<br />
*Average score: 61.22<br />
*AIME floor: 102<br />
*Distinction: 103.5 <br />
*Distinguished Honor Roll: 120<br />
<br />
===AMC 12A===<br />
*Average score: 61.42<br />
*AIME floor: 87<br />
*Distinction: 100.5<br />
*Distinguished Honor Roll: 123<br />
<br />
===AMC 12B===<br />
*Average score: 60.47<br />
*AIME floor: 87<br />
*Distinction: 97.5<br />
*Distinguished Honor Roll: 120<br />
<br />
===AIME I===<br />
*Average score: 5.70<br />
*Median score: 6<br />
*USOMO cutoff: 233.5 (AMC 12A), 235 (AMC 12B)<br />
*USOJMO cutoff: 229.5 (AMC 10A), 230 (AMC 10B)<br />
<br />
===AIME II===<br />
Due to COVID-19, the 2020 AIME II was administered online and referred to as the AOIME.<br />
*Average score: 6.1259<br />
*Median score: 6<br />
*USOMO cutoff: 234 (AMC 12A), 234.5 (AMC 12B)<br />
*USOJMO cutoff: 233.5 (AMC 10A), 229.5 (AMC 10B)<br />
<br />
===AMC 8===<br />
*Average score: 10.01<br />
*Honor Roll: 18<br />
*DHR: 21<br />
<br />
==2019==<br />
===AMC 10A===<br />
*Average score: 51.69<br />
*Honor roll: 96<br />
*AIME floor: 103.5<br />
*DHR: 123<br />
<br />
===AMC 10B===<br />
*Average score: 58.47<br />
*Honor roll: 102<br />
*AIME floor: 108<br />
*Distinguished Honor Roll: 121.5<br />
<br />
===AMC 12A===<br />
*Average score: 49.22<br />
*AIME floor: 84<br />
*DHR: 121.5<br />
<br />
===AMC 12B===<br />
*Average score: 56.74<br />
*AIME floor: 94.5 <br />
*DHR: 123<br />
<br />
===AIME I===<br />
*Average score: 5.88<br />
*Median score: 6<br />
*USAMO cutoff: 220 (AMC 12A), 230.5 (AMC 12B)<br />
*USAJMO cutoff: 209.5 (AMC 10A), 216 (AMC 10B)<br />
<br />
===AIME II===<br />
*Average score: 6.47<br />
*Median score: 6<br />
*USAMO cutoff: 230.5 (AMC 12A), 236 (AMC 12B)<br />
*USAJMO cutoff: 216.5 (AMC 10A), 220.5 (AMC 10B)<br />
<br />
===AMC 8===<br />
*Average score: 9.43<br />
*Honor roll: 19<br />
*DHR: 23<br />
<br />
==2018==<br />
===AMC 10A===<br />
*Average score: 53.84<br />
*Honor roll: 100.5<br />
*AIME floor: 111<br />
*DHR: 127.5<br />
<br />
===AMC 10B===<br />
*Average score: 57.81<br />
*Honor roll: 97.5<br />
*AIME floor: 108<br />
*DHR: 123<br />
<br />
===AMC 12A===<br />
*Average score: 56.36<br />
*AIME floor: 93<br />
*DHR: 120<br />
<br />
===AMC 12B===<br />
*Average score: 57.85<br />
*AIME floor: 99<br />
*DHR: 126<br />
<br />
===AIME I===<br />
*Average score: 5.09<br />
*Median score: 5<br />
*USAMO cutoff: 215 (AMC 12A), 235 (AMC 12B)<br />
*USAJMO cutoff: 222 (AMC 10A), 212 (AMC 10B)<br />
<br />
===AIME II===<br />
*Average score: 5.48<br />
*Median score: 5<br />
*USAMO cutoff: 216 (AMC 12A), 230.5 (AMC 12B)<br />
*USAJMO cutoff: 222 (AMC 10A), 212 (AMC 10B)<br />
<br />
===AMC 8===<br />
*Average score: 8.51<br />
*Honor roll: 15<br />
*DHR: 19<br />
<br />
==2017==<br />
===AMC 10A===<br />
*Average score: 59.33<br />
*AIME floor: 112.5<br />
*DHR: 127.5<br />
<br />
===AMC 10B===<br />
*Average score: 66.56<br />
*AIME floor: 120<br />
*DHR: 136.5<br />
<br />
===AMC 12A===<br />
*Average score: 60.32<br />
*AIME floor: 96<br />
*DHR: 115.5<br />
<br />
===AMC 12B===<br />
*Average score: 58.35<br />
*AIME floor: 100<br />
*DHR: 129<br />
<br />
===AIME I===<br />
*Average score: 5.69<br />
*Median score: 5<br />
*USAMO cutoff: 225 (AMC 12A), 235 (AMC 12B)<br />
*USAJMO cutoff: 224.5 (AMC 10A), 233 (AMC 10B)<br />
<br />
===AIME II===<br />
*Average score: 5.64<br />
*Median score: 5<br />
*USAMO cutoff: 221 (AMC 12A), 230.5 (AMC 12B)<br />
*USAJMO cutoff: 219 (AMC 10A), 225 (AMC 10B)<br />
<br />
===AMC 8===<br />
*Average score: 8.96<br />
*Honor roll: 17<br />
*DHR: 20<br />
<br />
==2016==<br />
===AMC 10A===<br />
*Average score: 65.31<br />
*AIME floor: 110<br />
*DHR: 120<br />
<br />
===AMC 10B===<br />
*Average score: 65.40<br />
*AIME floor: 110<br />
*DHR: 124.5<br />
<br />
===AMC 12A===<br />
*Average score: 59.06<br />
*AIME floor: 93<br />
*DHR: 111<br />
<br />
===AMC 12B===<br />
*Average score: 67.96<br />
*AIME floor: 100.5<br />
*DHR: 127.5<br />
<br />
===AIME I===<br />
*Average score: 5.83<br />
*Median score: 6<br />
*USAMO cutoff: 220<br />
*USAJMO cutoff: 210.5<br />
<br />
===AIME II===<br />
*Average score: 4.43<br />
*Median score: 4<br />
*USAMO cutoff: 205<br />
*USAJMO cutoff: 200<br />
<br />
===AMC 8===<br />
*Average score: 9.36<br />
*Honor roll: 18<br />
*DHR: 22<br />
<br />
==2015==<br />
===AMC 10A===<br />
*Average score: 73.39<br />
*AIME floor: 106.5<br />
*DHR: 115.5<br />
<br />
===AMC 10B===<br />
*Average score: 76.10<br />
*AIME floor: 120<br />
*DHR: 132<br />
<br />
===AMC 12A===<br />
*Average score: 69.90<br />
*AIME floor: 99<br />
*DHR: 117<br />
<br />
===AMC 12B===<br />
*Average score: 66.92<br />
*AIME floor: 100.5<br />
*DHR: 126<br />
<br />
===AIME I===<br />
*Average score: 5.29<br />
*Median score: 5<br />
*USAMO cutoff: 219.0<br />
*USAJMO cutoff: 213.0<br />
<br />
===AIME II===<br />
*Average score: 6.63<br />
*Median score: 6<br />
*USAMO cutoff: 229.0<br />
*USAJMO cutoff: 223.5<br />
<br />
===AMC 8===<br />
*Average score: 8.55<br />
*Honor roll: 16<br />
*DHR: 21<br />
<br />
==2014==<br />
===AMC 10A===<br />
*Average score: 63.83<br />
*AIME floor: 120<br />
*DHR: 132<br />
<br />
===AMC 10B===<br />
*Average score: 71.44<br />
*AIME floor: 120<br />
*DHR: 132<br />
<br />
===AMC 12A===<br />
*Average score: 64.01<br />
*AIME floor: 93<br />
*DHR: 109.5<br />
<br />
===AMC 12B===<br />
*Average score: 68.11<br />
*AIME floor: 100.5<br />
*DHR: 121.5<br />
<br />
===AIME I===<br />
*Average score: 4.88<br />
*Median score: 5<br />
*USAMO cutoff: 211.5<br />
*USAJMO cutoff: 211<br />
<br />
===AIME II===<br />
*Average score: 5.49<br />
*Median score: 5<br />
*USAMO cutoff: 211.5<br />
*USAJMO cutoff: 211<br />
<br />
===AMC 8===<br />
*Average score: 11.43<br />
*Honor roll: 19<br />
*DHR: 23<br />
<br />
==2013==<br />
===AMC 10A===<br />
*Average score: 72.50<br />
*AIME floor: 108<br />
*DHR: 117<br />
<br />
===AMC 10B===<br />
*Average score: 72.62<br />
*AIME floor: 120<br />
*DHR: 129<br />
<br />
===AMC 12A===<br />
*Average score: 65.06<br />
*AIME floor: 88.5<br />
*DHR: 106.5<br />
<br />
===AMC 12B===<br />
*Average score: 64.21<br />
*AIME floor: 93<br />
*DHR: 108<br />
<br />
===AIME I===<br />
*Average score: 4.69<br />
*Median score: 4<br />
*USAMO cutoff: 209<br />
*USAJMO cutoff: 210.5<br />
<br />
===AIME II===<br />
*Average score: 6.56<br />
*Median score: 6<br />
*USAMO cutoff: 209<br />
*USAJMO cutoff: 210.5<br />
<br />
===AMC 8===<br />
*Average score: 10.69<br />
*Honor roll: 18<br />
*DHR: 22<br />
<br />
==2012==<br />
===AMC 10A===<br />
*Average score: 72.51<br />
*AIME floor: 115.5<br />
*DHR: 121.5<br />
<br />
===AMC 10B===<br />
*Average score: 76.59<br />
*AIME floor: 120<br />
*DHR: 133.5<br />
<br />
===AMC 12A===<br />
*Average score: 64.62<br />
*AIME floor: 94.5<br />
*DHR: 109.5<br />
<br />
===AMC 12B===<br />
*Average score: 70.08<br />
*AIME floor: 99<br />
*DHR: 114<br />
<br />
===AIME I===<br />
*Average score: 5.13<br />
*Median score: 5<br />
*USAMO cutoff: 204.5<br />
*USAJMO cutoff: 204<br />
<br />
===AIME II===<br />
*Average score: 4.94<br />
*Median score: 5<br />
*USAMO cutoff: 204.5<br />
*USAJMO cutoff: 204<br />
<br />
===AMC 8===<br />
*Average score: 10.67<br />
*Honor roll: 18<br />
*DHR: 22<br />
<br />
==2011==<br />
===AMC 10A===<br />
*Average score: 64.24<br />
*AIME floor: 117<br />
*DHR: 129<br />
<br />
===AMC 10B===<br />
*Average score: 71.78<br />
*AIME floor: 117<br />
*DHR: 133.5<br />
<br />
===AMC 12A===<br />
*Average score: 65.38<br />
*AIME floor: 93<br />
*DHR: 112.5<br />
<br />
===AMC 12B===<br />
*Average score: 64.71<br />
*AIME floor: 97.5<br />
*DHR: 121.5<br />
<br />
===AIME I===<br />
*Average score: 2.23<br />
*Median score: 2<br />
*USAMO cutoff: 188<br />
*USAJMO cutoff: 179<br />
<br />
===AIME II===<br />
*Average score: 5.47<br />
*Median score: 5<br />
*USAMO cutoff: 215.5<br />
*USAJMO cutoff: 196.5<br />
<br />
===AMC 8===<br />
*Average score: 10.75<br />
*Honor roll: 17<br />
*DHR: 22<br />
<br />
==2010==<br />
===AMC 10A===<br />
*Average score: 68.11<br />
*AIME floor: 115.5<br />
<br />
===AMC 10B===<br />
*Average score: 68.57<br />
*AIME floor: 118.5<br />
<br />
===AMC 12A===<br />
*Average score: 61.02<br />
*AIME floor: 88.5<br />
*DHR cutoff: 108<br />
<br />
===AMC 12B===<br />
*Average score: 59.58<br />
*AIME floor: 88.5<br />
*DHR cutoff: 109.5<br />
<br />
===AIME I===<br />
*Average score: 5.90<br />
*Median score: 6<br />
*USAMO cutoff: 208.5 (204.5 for non juniors and seniors)<br />
*USAJMO cutoff: 188.5<br />
<br />
===AIME II===<br />
*Average score: 3.39<br />
*Median score: 3<br />
*USAMO cutoff: 208.5 (204.5 for non juniors and seniors)<br />
*USAJMO cutoff: 188.5<br />
<br />
===AMC 8===<br />
*Average score: 9.59<br />
*Honor roll: 17<br />
*DHR: 22<br />
<br />
==2009==<br />
===AMC 10A===<br />
*Average score: 67.41<br />
*AIME floor: 117<br />
<br />
===AMC 10B===<br />
*Average score: 74.73<br />
*AIME floor: 120<br />
<br />
===AMC 12A===<br />
*Average score: 66.37<br />
*AIME floor: 97.5<br />
<br />
===AMC 12B===<br />
*Average score: 71.88<br />
*AIME floor: 100 (Top 5% (1.00))<br />
<br />
===AIME I===<br />
*Average score: 4.17<br />
*Median score: 4<br />
*USAMO floor: <br />
<br />
===AIME II===<br />
*Average score: 3.27<br />
*Median score: 3<br />
*USAMO floor:<br />
<br />
===AMC 8===<br />
*Average score: 10.28<br />
*Honor roll: 17<br />
*DHR: 20<br />
<br />
==2008==<br />
===AMC 10A===<br />
*Average score: 60.25<br />
*AIME floor: 117<br />
<br />
===AMC 10B===<br />
*Average score: <br />
*AIME floor: 120<br />
<br />
===AMC 12A===<br />
*Average score: 65.6<br />
*AIME floor: 97.5<br />
<br />
===AMC 12B===<br />
*Average score: 68.9<br />
*AIME floor: 97.5<br />
<br />
===AIME I===<br />
*Average score: 4.77<br />
*Median score: 4<br />
*USAMO floor: <br />
<br />
===AIME II===<br />
*Average score: 5.27<br />
*Median score: 5<br />
*USAMO floor:<br />
<br />
===AMC 8===<br />
*Average score: 11.45<br />
*Honor roll: 19<br />
*DHR: 22<br />
<br />
==2007==<br />
<br />
===AMC 10A===<br />
*Average score: 67.9<br />
*AIME floor: 117<br />
<br />
===AMC 10B=== <br />
*Average score: 61.5<br />
*AIME floor: 115.5<br />
<br />
===AMC 12A===<br />
*Average score: 66.8<br />
*AIME floor: 97.5<br />
<br />
===AMC 12B===<br />
*Average score: 73.1<br />
*AIME floor: 100<br />
<br />
===AIME I===<br />
*Average score: 5<br />
*Median score: 3<br />
*USAMO floor: 6<br />
<br />
===AIME II===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
===AMC 8===<br />
*Average score: 9.87<br />
*Honor roll: 17<br />
*DHR: 21<br />
<br />
==2006==<br />
===AMC 10A===<br />
*Average score: 79.0<br />
*AIME floor: 120<br />
<br />
===AMC 10B===<br />
*Average score: 68.5<br />
*AIME floor: 120<br />
<br />
===AMC 12A===<br />
*Average score: 85.7<br />
*AIME floor: 100<br />
<br />
===AMC 12B===<br />
*Average score: 85.5<br />
*AIME floor: 100<br />
<br />
===AIME I===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
===AIME II===<br />
*Average score: <br />
*Median score:<br />
*USAMO floor:<br />
<br />
===AMC 8===<br />
*Honor roll: 17<br />
*DHR: 21<br />
<br />
==2005==<br />
===AMC 10A===<br />
*Average score: 74.0<br />
*AIME floor: 120<br />
<br />
===AMC 10B===<br />
*Average score: 79.0<br />
*AIME floor: 120<br />
<br />
===AMC 12A===<br />
*Average score: 78.7<br />
*AIME floor: 100<br />
<br />
===AMC 12B===<br />
*Average score: 83.4<br />
*AIME floor: 100<br />
<br />
===AIME I===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
===AIME II===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
===AMC 8===<br />
*Honor roll: 16<br />
*DHR: 20<br />
<br />
==2004==<br />
===AMC 10A===<br />
*Average score: 69.1<br />
*AIME floor: 110<br />
<br />
===AMC 10B===<br />
*Average score: 80.4<br />
*AIME floor: 115<br />
<br />
===AMC 12A===<br />
*Average score: 73.9<br />
*AIME floor: 100<br />
<br />
===AMC 12B===<br />
*Average score: 84.5<br />
*AIME floor: 100<br />
<br />
===AIME I===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
===AIME II===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
===AMC 8===<br />
*Honor roll: 17<br />
*DHR: 21<br />
<br />
==2003==<br />
===AMC 10A===<br />
*Average score: 74.4<br />
*AIME floor: 119<br />
<br />
===AMC 10B===<br />
*Average score: 79.6<br />
*AIME floor: 121<br />
<br />
===AMC 12A===<br />
*Average score: 77.8<br />
*AIME floor: 100<br />
<br />
===AMC 12B===<br />
*Average score: 76.6<br />
*AIME floor: 100<br />
<br />
===AIME I===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
===AIME II===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
===AMC 8===<br />
*Honor roll: 18<br />
*DHR: 22<br />
<br />
==2002==<br />
===AMC 10A===<br />
*Average score: 68.5<br />
*AIME floor: 115<br />
<br />
===AMC 10B===<br />
*Average score: 74.9<br />
*AIME floor: 118<br />
<br />
===AMC 12A===<br />
*Average score: 72.7<br />
*AIME floor: 100<br />
<br />
===AMC 12B===<br />
*Average score: 80.8<br />
*AIME floor: 100<br />
<br />
===AIME I===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
===AIME II===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
==2001==<br />
===AMC 10===<br />
*Average score: 67.8<br />
*AIME floor: 116<br />
<br />
===AMC 12===<br />
*Average score: 56.6<br />
*AIME floor: 84<br />
<br />
===AIME I===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
===AIME II===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
==2000==<br />
===AMC 10===<br />
*Average score: <math>64.2</math><br />
*AIME floor: <math>110</math><br />
<br />
===AMC 12===<br />
*Average score: <math>64.9</math><br />
*AIME floor: <math>92</math><br />
<br />
===AIME I===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
===AIME II===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
==1999==<br />
===AHSME===<br />
*Average score: <math>68.8</math><br />
*AIME floor:<br />
<br />
===AIME===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:</div>Theajlhttps://artofproblemsolving.com/wiki/index.php?title=AMC_historical_results&diff=146349AMC historical results2021-02-13T03:48:58Z<p>Theajl: /* AMC 10A */</p>
<hr />
<div><!-- Post AMC statistics and lists of high scorers here so that the AMC page doesn't get cluttered. --><br />
This is the '''AMC historical results''' page. This page should include results for the [[AIME]] as well. For [[USAMO]] results, see [[USAMO historical results]].<br />
==2021==<br />
===AMC 10A===<br />
*Average score: <br />
*AIME floor:<br />
*Distinction:<br />
*Distinguished Honor Roll:<br />
<br />
===AMC 10B===<br />
*Average score: 0<br />
*AIME floor: -1.5 (Expanded due to cheating concerns.)<br />
*Distinction: -1<br />
*Distinguished Honor Roll: 0.2<br />
===AMC 12A===<br />
*Average score: 36<br />
*AIME floor: 37.5 (Expanded due to cheating concerns.)<br />
*Distinction: 42<br />
*Distinguished Honor Roll: 43<br />
<br />
===AMC 12B===<br />
*Average score: 33.3333333<br />
*AIME floor: 69 (Expanded due to cheating concerns.)<br />
*Distinction: 4<br />
*Distinguished Honor Roll: 20<br />
===AIME I===<br />
*Average score: <br />
*Median score: <br />
*USAMO cutoff: <br />
*USAJMO cutoff:<br />
===AIME II===<br />
*Average score: <br />
*Median score: <br />
*USAMO cutoff: <br />
*USAJMO cutoff:<br />
===AMC 8===<br />
*Average score:<br />
*Honor Roll: <br />
*DHR:<br />
<br />
==2020==<br />
===AMC 10A===<br />
*Average score: 64.29<br />
*AIME floor: 103.5<br />
*Distinction: 105<br />
*Distinguished Honor Roll: 124.5<br />
<br />
===AMC 10B===<br />
*Average score: 61.22<br />
*AIME floor: 102<br />
*Distinction: 103.5 <br />
*Distinguished Honor Roll: 120<br />
<br />
===AMC 12A===<br />
*Average score: 61.42<br />
*AIME floor: 87<br />
*Distinction: 100.5<br />
*Distinguished Honor Roll: 123<br />
<br />
===AMC 12B===<br />
*Average score: 60.47<br />
*AIME floor: 87<br />
*Distinction: 97.5<br />
*Distinguished Honor Roll: 120<br />
<br />
===AIME I===<br />
*Average score: 5.70<br />
*Median score: 6<br />
*USOMO cutoff: 233.5 (AMC 12A), 235 (AMC 12B)<br />
*USOJMO cutoff: 229.5 (AMC 10A), 230 (AMC 10B)<br />
<br />
===AIME II===<br />
Due to COVID-19, the 2020 AIME II was administered online and referred to as the AOIME.<br />
*Average score: 6.1259<br />
*Median score: 6<br />
*USOMO cutoff: 234 (AMC 12A), 234.5 (AMC 12B)<br />
*USOJMO cutoff: 233.5 (AMC 10A), 229.5 (AMC 10B)<br />
<br />
===AMC 8===<br />
*Average score: 10.01<br />
*Honor Roll: 18<br />
*DHR: 21<br />
<br />
==2019==<br />
===AMC 10A===<br />
*Average score: 51.69<br />
*Honor roll: 96<br />
*AIME floor: 103.5<br />
*DHR: 123<br />
<br />
===AMC 10B===<br />
*Average score: 58.47<br />
*Honor roll: 102<br />
*AIME floor: 108<br />
*Distinguished Honor Roll: 121.5<br />
<br />
===AMC 12A===<br />
*Average score: 49.22<br />
*AIME floor: 84<br />
*DHR: 121.5<br />
<br />
===AMC 12B===<br />
*Average score: 56.74<br />
*AIME floor: 94.5 <br />
*DHR: 123<br />
<br />
===AIME I===<br />
*Average score: 5.88<br />
*Median score: 6<br />
*USAMO cutoff: 220 (AMC 12A), 230.5 (AMC 12B)<br />
*USAJMO cutoff: 209.5 (AMC 10A), 216 (AMC 10B)<br />
<br />
===AIME II===<br />
*Average score: 6.47<br />
*Median score: 6<br />
*USAMO cutoff: 230.5 (AMC 12A), 236 (AMC 12B)<br />
*USAJMO cutoff: 216.5 (AMC 10A), 220.5 (AMC 10B)<br />
<br />
===AMC 8===<br />
*Average score: 9.43<br />
*Honor roll: 19<br />
*DHR: 23<br />
<br />
==2018==<br />
===AMC 10A===<br />
*Average score: 53.84<br />
*Honor roll: 100.5<br />
*AIME floor: 111<br />
*DHR: 127.5<br />
<br />
===AMC 10B===<br />
*Average score: 57.81<br />
*Honor roll: 97.5<br />
*AIME floor: 108<br />
*DHR: 123<br />
<br />
===AMC 12A===<br />
*Average score: 56.36<br />
*AIME floor: 93<br />
*DHR: 120<br />
<br />
===AMC 12B===<br />
*Average score: 57.85<br />
*AIME floor: 99<br />
*DHR: 126<br />
<br />
===AIME I===<br />
*Average score: 5.09<br />
*Median score: 5<br />
*USAMO cutoff: 215 (AMC 12A), 235 (AMC 12B)<br />
*USAJMO cutoff: 222 (AMC 10A), 212 (AMC 10B)<br />
<br />
===AIME II===<br />
*Average score: 5.48<br />
*Median score: 5<br />
*USAMO cutoff: 216 (AMC 12A), 230.5 (AMC 12B)<br />
*USAJMO cutoff: 222 (AMC 10A), 212 (AMC 10B)<br />
<br />
===AMC 8===<br />
*Average score: 8.51<br />
*Honor roll: 15<br />
*DHR: 19<br />
<br />
==2017==<br />
===AMC 10A===<br />
*Average score: 59.33<br />
*AIME floor: 112.5<br />
*DHR: 127.5<br />
<br />
===AMC 10B===<br />
*Average score: 66.56<br />
*AIME floor: 120<br />
*DHR: 136.5<br />
<br />
===AMC 12A===<br />
*Average score: 60.32<br />
*AIME floor: 96<br />
*DHR: 115.5<br />
<br />
===AMC 12B===<br />
*Average score: 58.35<br />
*AIME floor: 100<br />
*DHR: 129<br />
<br />
===AIME I===<br />
*Average score: 5.69<br />
*Median score: 5<br />
*USAMO cutoff: 225 (AMC 12A), 235 (AMC 12B)<br />
*USAJMO cutoff: 224.5 (AMC 10A), 233 (AMC 10B)<br />
<br />
===AIME II===<br />
*Average score: 5.64<br />
*Median score: 5<br />
*USAMO cutoff: 221 (AMC 12A), 230.5 (AMC 12B)<br />
*USAJMO cutoff: 219 (AMC 10A), 225 (AMC 10B)<br />
<br />
===AMC 8===<br />
*Average score: 8.96<br />
*Honor roll: 17<br />
*DHR: 20<br />
<br />
==2016==<br />
===AMC 10A===<br />
*Average score: 65.31<br />
*AIME floor: 110<br />
*DHR: 120<br />
<br />
===AMC 10B===<br />
*Average score: 65.40<br />
*AIME floor: 110<br />
*DHR: 124.5<br />
<br />
===AMC 12A===<br />
*Average score: 59.06<br />
*AIME floor: 93<br />
*DHR: 111<br />
<br />
===AMC 12B===<br />
*Average score: 67.96<br />
*AIME floor: 100.5<br />
*DHR: 127.5<br />
<br />
===AIME I===<br />
*Average score: 5.83<br />
*Median score: 6<br />
*USAMO cutoff: 220<br />
*USAJMO cutoff: 210.5<br />
<br />
===AIME II===<br />
*Average score: 4.43<br />
*Median score: 4<br />
*USAMO cutoff: 205<br />
*USAJMO cutoff: 200<br />
<br />
===AMC 8===<br />
*Average score: 9.36<br />
*Honor roll: 18<br />
*DHR: 22<br />
<br />
==2015==<br />
===AMC 10A===<br />
*Average score: 73.39<br />
*AIME floor: 106.5<br />
*DHR: 115.5<br />
<br />
===AMC 10B===<br />
*Average score: 76.10<br />
*AIME floor: 120<br />
*DHR: 132<br />
<br />
===AMC 12A===<br />
*Average score: 69.90<br />
*AIME floor: 99<br />
*DHR: 117<br />
<br />
===AMC 12B===<br />
*Average score: 66.92<br />
*AIME floor: 100.5<br />
*DHR: 126<br />
<br />
===AIME I===<br />
*Average score: 5.29<br />
*Median score: 5<br />
*USAMO cutoff: 219.0<br />
*USAJMO cutoff: 213.0<br />
<br />
===AIME II===<br />
*Average score: 6.63<br />
*Median score: 6<br />
*USAMO cutoff: 229.0<br />
*USAJMO cutoff: 223.5<br />
<br />
===AMC 8===<br />
*Average score: 8.55<br />
*Honor roll: 16<br />
*DHR: 21<br />
<br />
==2014==<br />
===AMC 10A===<br />
*Average score: 63.83<br />
*AIME floor: 120<br />
*DHR: 132<br />
<br />
===AMC 10B===<br />
*Average score: 71.44<br />
*AIME floor: 120<br />
*DHR: 132<br />
<br />
===AMC 12A===<br />
*Average score: 64.01<br />
*AIME floor: 93<br />
*DHR: 109.5<br />
<br />
===AMC 12B===<br />
*Average score: 68.11<br />
*AIME floor: 100.5<br />
*DHR: 121.5<br />
<br />
===AIME I===<br />
*Average score: 4.88<br />
*Median score: 5<br />
*USAMO cutoff: 211.5<br />
*USAJMO cutoff: 211<br />
<br />
===AIME II===<br />
*Average score: 5.49<br />
*Median score: 5<br />
*USAMO cutoff: 211.5<br />
*USAJMO cutoff: 211<br />
<br />
===AMC 8===<br />
*Average score: 11.43<br />
*Honor roll: 19<br />
*DHR: 23<br />
<br />
==2013==<br />
===AMC 10A===<br />
*Average score: 72.50<br />
*AIME floor: 108<br />
*DHR: 117<br />
<br />
===AMC 10B===<br />
*Average score: 72.62<br />
*AIME floor: 120<br />
*DHR: 129<br />
<br />
===AMC 12A===<br />
*Average score: 65.06<br />
*AIME floor: 88.5<br />
*DHR: 106.5<br />
<br />
===AMC 12B===<br />
*Average score: 64.21<br />
*AIME floor: 93<br />
*DHR: 108<br />
<br />
===AIME I===<br />
*Average score: 4.69<br />
*Median score: 4<br />
*USAMO cutoff: 209<br />
*USAJMO cutoff: 210.5<br />
<br />
===AIME II===<br />
*Average score: 6.56<br />
*Median score: 6<br />
*USAMO cutoff: 209<br />
*USAJMO cutoff: 210.5<br />
<br />
===AMC 8===<br />
*Average score: 10.69<br />
*Honor roll: 18<br />
*DHR: 22<br />
<br />
==2012==<br />
===AMC 10A===<br />
*Average score: 72.51<br />
*AIME floor: 115.5<br />
*DHR: 121.5<br />
<br />
===AMC 10B===<br />
*Average score: 76.59<br />
*AIME floor: 120<br />
*DHR: 133.5<br />
<br />
===AMC 12A===<br />
*Average score: 64.62<br />
*AIME floor: 94.5<br />
*DHR: 109.5<br />
<br />
===AMC 12B===<br />
*Average score: 70.08<br />
*AIME floor: 99<br />
*DHR: 114<br />
<br />
===AIME I===<br />
*Average score: 5.13<br />
*Median score: 5<br />
*USAMO cutoff: 204.5<br />
*USAJMO cutoff: 204<br />
<br />
===AIME II===<br />
*Average score: 4.94<br />
*Median score: 5<br />
*USAMO cutoff: 204.5<br />
*USAJMO cutoff: 204<br />
<br />
===AMC 8===<br />
*Average score: 10.67<br />
*Honor roll: 18<br />
*DHR: 22<br />
<br />
==2011==<br />
===AMC 10A===<br />
*Average score: 64.24<br />
*AIME floor: 117<br />
*DHR: 129<br />
<br />
===AMC 10B===<br />
*Average score: 71.78<br />
*AIME floor: 117<br />
*DHR: 133.5<br />
<br />
===AMC 12A===<br />
*Average score: 65.38<br />
*AIME floor: 93<br />
*DHR: 112.5<br />
<br />
===AMC 12B===<br />
*Average score: 64.71<br />
*AIME floor: 97.5<br />
*DHR: 121.5<br />
<br />
===AIME I===<br />
*Average score: 2.23<br />
*Median score: 2<br />
*USAMO cutoff: 188<br />
*USAJMO cutoff: 179<br />
<br />
===AIME II===<br />
*Average score: 5.47<br />
*Median score: 5<br />
*USAMO cutoff: 215.5<br />
*USAJMO cutoff: 196.5<br />
<br />
===AMC 8===<br />
*Average score: 10.75<br />
*Honor roll: 17<br />
*DHR: 22<br />
<br />
==2010==<br />
===AMC 10A===<br />
*Average score: 68.11<br />
*AIME floor: 115.5<br />
<br />
===AMC 10B===<br />
*Average score: 68.57<br />
*AIME floor: 118.5<br />
<br />
===AMC 12A===<br />
*Average score: 61.02<br />
*AIME floor: 88.5<br />
*DHR cutoff: 108<br />
<br />
===AMC 12B===<br />
*Average score: 59.58<br />
*AIME floor: 88.5<br />
*DHR cutoff: 109.5<br />
<br />
===AIME I===<br />
*Average score: 5.90<br />
*Median score: 6<br />
*USAMO cutoff: 208.5 (204.5 for non juniors and seniors)<br />
*USAJMO cutoff: 188.5<br />
<br />
===AIME II===<br />
*Average score: 3.39<br />
*Median score: 3<br />
*USAMO cutoff: 208.5 (204.5 for non juniors and seniors)<br />
*USAJMO cutoff: 188.5<br />
<br />
===AMC 8===<br />
*Average score: 9.59<br />
*Honor roll: 17<br />
*DHR: 22<br />
<br />
==2009==<br />
===AMC 10A===<br />
*Average score: 67.41<br />
*AIME floor: 117<br />
<br />
===AMC 10B===<br />
*Average score: 74.73<br />
*AIME floor: 120<br />
<br />
===AMC 12A===<br />
*Average score: 66.37<br />
*AIME floor: 97.5<br />
<br />
===AMC 12B===<br />
*Average score: 71.88<br />
*AIME floor: 100 (Top 5% (1.00))<br />
<br />
===AIME I===<br />
*Average score: 4.17<br />
*Median score: 4<br />
*USAMO floor: <br />
<br />
===AIME II===<br />
*Average score: 3.27<br />
*Median score: 3<br />
*USAMO floor:<br />
<br />
===AMC 8===<br />
*Average score: 10.28<br />
*Honor roll: 17<br />
*DHR: 20<br />
<br />
==2008==<br />
===AMC 10A===<br />
*Average score: 60.25<br />
*AIME floor: 117<br />
<br />
===AMC 10B===<br />
*Average score: <br />
*AIME floor: 120<br />
<br />
===AMC 12A===<br />
*Average score: 65.6<br />
*AIME floor: 97.5<br />
<br />
===AMC 12B===<br />
*Average score: 68.9<br />
*AIME floor: 97.5<br />
<br />
===AIME I===<br />
*Average score: 4.77<br />
*Median score: 4<br />
*USAMO floor: <br />
<br />
===AIME II===<br />
*Average score: 5.27<br />
*Median score: 5<br />
*USAMO floor:<br />
<br />
===AMC 8===<br />
*Average score: 11.45<br />
*Honor roll: 19<br />
*DHR: 22<br />
<br />
==2007==<br />
<br />
===AMC 10A===<br />
*Average score: 67.9<br />
*AIME floor: 117<br />
<br />
===AMC 10B=== <br />
*Average score: 61.5<br />
*AIME floor: 115.5<br />
<br />
===AMC 12A===<br />
*Average score: 66.8<br />
*AIME floor: 97.5<br />
<br />
===AMC 12B===<br />
*Average score: 73.1<br />
*AIME floor: 100<br />
<br />
===AIME I===<br />
*Average score: 5<br />
*Median score: 3<br />
*USAMO floor: 6<br />
<br />
===AIME II===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
===AMC 8===<br />
*Average score: 9.87<br />
*Honor roll: 17<br />
*DHR: 21<br />
<br />
==2006==<br />
===AMC 10A===<br />
*Average score: 79.0<br />
*AIME floor: 120<br />
<br />
===AMC 10B===<br />
*Average score: 68.5<br />
*AIME floor: 120<br />
<br />
===AMC 12A===<br />
*Average score: 85.7<br />
*AIME floor: 100<br />
<br />
===AMC 12B===<br />
*Average score: 85.5<br />
*AIME floor: 100<br />
<br />
===AIME I===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
===AIME II===<br />
*Average score: <br />
*Median score:<br />
*USAMO floor:<br />
<br />
===AMC 8===<br />
*Honor roll: 17<br />
*DHR: 21<br />
<br />
==2005==<br />
===AMC 10A===<br />
*Average score: 74.0<br />
*AIME floor: 120<br />
<br />
===AMC 10B===<br />
*Average score: 79.0<br />
*AIME floor: 120<br />
<br />
===AMC 12A===<br />
*Average score: 78.7<br />
*AIME floor: 100<br />
<br />
===AMC 12B===<br />
*Average score: 83.4<br />
*AIME floor: 100<br />
<br />
===AIME I===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
===AIME II===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
===AMC 8===<br />
*Honor roll: 16<br />
*DHR: 20<br />
<br />
==2004==<br />
===AMC 10A===<br />
*Average score: 69.1<br />
*AIME floor: 110<br />
<br />
===AMC 10B===<br />
*Average score: 80.4<br />
*AIME floor: 115<br />
<br />
===AMC 12A===<br />
*Average score: 73.9<br />
*AIME floor: 100<br />
<br />
===AMC 12B===<br />
*Average score: 84.5<br />
*AIME floor: 100<br />
<br />
===AIME I===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
===AIME II===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
===AMC 8===<br />
*Honor roll: 17<br />
*DHR: 21<br />
<br />
==2003==<br />
===AMC 10A===<br />
*Average score: 74.4<br />
*AIME floor: 119<br />
<br />
===AMC 10B===<br />
*Average score: 79.6<br />
*AIME floor: 121<br />
<br />
===AMC 12A===<br />
*Average score: 77.8<br />
*AIME floor: 100<br />
<br />
===AMC 12B===<br />
*Average score: 76.6<br />
*AIME floor: 100<br />
<br />
===AIME I===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
===AIME II===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
===AMC 8===<br />
*Honor roll: 18<br />
*DHR: 22<br />
<br />
==2002==<br />
===AMC 10A===<br />
*Average score: 68.5<br />
*AIME floor: 115<br />
<br />
===AMC 10B===<br />
*Average score: 74.9<br />
*AIME floor: 118<br />
<br />
===AMC 12A===<br />
*Average score: 72.7<br />
*AIME floor: 100<br />
<br />
===AMC 12B===<br />
*Average score: 80.8<br />
*AIME floor: 100<br />
<br />
===AIME I===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
===AIME II===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
==2001==<br />
===AMC 10===<br />
*Average score: 67.8<br />
*AIME floor: 116<br />
<br />
===AMC 12===<br />
*Average score: 56.6<br />
*AIME floor: 84<br />
<br />
===AIME I===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
===AIME II===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
==2000==<br />
===AMC 10===<br />
*Average score: <math>64.2</math><br />
*AIME floor: <math>110</math><br />
<br />
===AMC 12===<br />
*Average score: <math>64.9</math><br />
*AIME floor: <math>92</math><br />
<br />
===AIME I===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
===AIME II===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:<br />
<br />
==1999==<br />
===AHSME===<br />
*Average score: <math>68.8</math><br />
*AIME floor:<br />
<br />
===AIME===<br />
*Average score:<br />
*Median score:<br />
*USAMO floor:</div>Theajlhttps://artofproblemsolving.com/wiki/index.php?title=2021_AMC_12B_Problems/Problem_25&diff=1460872021 AMC 12B Problems/Problem 252021-02-12T17:41:54Z<p>Theajl: /* Solution */</p>
<hr />
<div>==Problem==<br />
<br />
Let <math>S</math> be the set of lattice points in the coordinate plane, both of whose coordinates are integers between <math>1</math> and <math>30,</math> inclusive. Exactly <math>300</math> points in <math>S</math> lie on or below a line with equation <math>y=mx.</math> The possible values of <math>m</math> lie in an interval of length <math>\frac ab,</math> where <math>a</math> and <math>b</math> are relatively prime positive integers. What is <math>a+b?</math><br />
<br />
<math>\textbf{(A)} ~31 \qquad \textbf{(B)} ~47 \qquad \textbf{(C)} ~62\qquad \textbf{(D)} ~72 \qquad \textbf{(E)} ~85</math><br />
<br />
==Solution==<br />
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 900 lattice points. I estimate that the slope, m, 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 600 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 are is above the line. <br />
<br />
I find that the number of coordinates with <math>x=1</math> above the line is 30, 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 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 <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 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. <br />
<br />
To find the upper bound, notice that each point with an integer-valued x-coordinate is either <math>\frac{1}{3}</math> or <math>\frac{2}{3}</math> 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 <math>x=28, 29, 30</math> which the line <math>y=\frac{2}{3}x</math> intersects at <math>y= \frac{56}{3}, \frac{58}{3}, 20</math>. 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, <math>\frac{56}{3}</math>) is closer to the lattice point. The slope of the line which goes through both the origin and (28,19) is <math>y=\frac{19}{28}x</math>. This gives an upper bound of <math>m=\frac{19}{28}</math>. <br />
<br />
Taking the upper bound of m and subtracting the lower bound yields <math>\frac{19}{28}-\frac{2}{3}=\frac{1}{84}</math>. This is answer <math>1+84=</math> <math>\boxed{\textbf{(E)} ~85}</math>.<br />
<br />
==See also==<br />
{{AMC12 box|year=2021|ab=B|num-b=24|after=Last problem}}<br />
<br />
[[Category:Intermediate Geometry Problems]]<br />
{{AMC10 box|year=2021|ab=B|num-b=24|after=Last problem}}</div>Theajlhttps://artofproblemsolving.com/wiki/index.php?title=2021_AMC_12B_Problems/Problem_25&diff=1460532021 AMC 12B Problems/Problem 252021-02-12T16:56:13Z<p>Theajl: /* Problem */</p>
<hr />
<div>==Problem==<br />
<br />
Let <math>S</math> be the set of lattice points in the coordinate plane, both of whose coordinates are integers between <math>1</math> and <math>30,</math> inclusive. Exactly <math>300</math> points in <math>S</math> lie on or below a line with equation <math>y=mx.</math> The possible values of <math>m</math> lie in an interval of length <math>\frac ab,</math> where <math>a</math> and <math>b</math> are relatively prime positive integers. What is <math>a+b?</math><br />
<br />
<math>\textbf{(A) }31 \qquad \textbf{(B) }47 \qquad \textbf{(C) }62\qquad \textbf{(D) }72 \qquad \textbf{(E) }85</math><br />
<br />
{{AMC10 box|year=2021|ab=B|num-b=24|after=Last Problem}}<br />
<br />
==Solution==<br />
<br />
<br />
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 900 lattice points. I estimate that the slope, m, 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 600 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 are is above the line. <br />
<br />
I find that the number of coordinates with <math>x=1</math> above the line is 30, 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 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 <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 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. <br />
<br />
To find the upper bound, notice that each point with an integer-valued x-coordinate is either <math>\frac{1}{3}</math> or <math>\frac{2}{3}</math> 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 <math>x=28, 29, 30</math> which the line <math>y=\frac{2}{3}x</math> intersects at <math>y= \frac{56}{3}, \frac{58}{3}, 20</math>. 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, <math>\frac{56}{3}</math>) is closer to the lattice point. The slope of the line which goes through both the origin and (28,19) is <math>y=\frac{19}{28}x</math>. This gives an upper bound of <math>m=\frac{19}{28}</math>. <br />
<br />
Taking the upper bound of m and subtracting the lower bound yields <math>\frac{19}{28}-\frac{2}{3}=\frac{1}{84}</math>. This is answer E.<br />
<br />
<br />
~Theajl</div>Theajlhttps://artofproblemsolving.com/wiki/index.php?title=2021_AMC_12B_Problems&diff=1457472021 AMC 12B Problems2021-02-12T04:26:42Z<p>Theajl: /* Problem 25 */</p>
<hr />
<div>{{AMC12 Problems|year=2021|ab=B}}<br />
==Problem 1==<br />
How many integer values of <math>x</math> satisfy <math>|x|<3\pi?</math><br />
<br />
<math>\textbf{(A) }9 \qquad \textbf{(B) }10 \qquad \textbf{(C) }18 \qquad \textbf{(D) }19 \qquad \textbf{(E) }20</math><br />
<br />
[[2021 AMC 12B Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
At a math contest, <math>57</math> students are wearing blue shirts, and another <math>75</math> students are wearing yellow shirts. The <math>132</math> students are assigned into <math>66</math> points. In exactly <math>23</math> of these pairs, both students are wearing blue shirts. In how many pairs are both studets wearing yellow shirts?<br />
<br />
<math>\textbf{(A) }23 \qquad \textbf{(B) }32 \qquad \textbf{(C) }37 \qquad \textbf{(D) }41 \qquad \textbf{(E) }64</math><br />
<br />
[[2021 AMC 12B Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
Suppose<cmath>2+\frac{1}{1+\frac{1}{2+\frac{2}{3+x}}}=\frac{144}{53}.</cmath>What is the value of <math>x?</math><br />
<br />
<math>\textbf{(A) }\frac34 \qquad \textbf{(B) }\frac78 \qquad \textbf{(C) }\frac{14}{15} \qquad \textbf{(D) }\frac{37}{38} \qquad \textbf{(E) }\frac{52}{53}</math><br />
<br />
[[2021 AMC 12B Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
Ms. Blackwell gives an exam to two classes. The mean of the scores of the students in the morning class is <math>84</math>, and the afternoon class's mean score is <math>70</math>. The ratio of the number of students in the morning clas to the number of students in the afternoon class is <math>\frac34</math>. What is the mean of the score of all the students?<br />
<br />
<math>\textbf{(A) }74 \qquad \textbf{(B) }75 \qquad \textbf{(C) }76 \qquad \textbf{(D) }77 \qquad \textbf{(E) }78</math><br />
<br />
[[2021 AMC 12B Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
The point <math>P(a,b)</math> in the <math>xy</math>-plane is first rotated counterclockwise by <math>90^\circ</math> around the point <math>(1,5)</math> and then reflected about the line <math>y=-x</math>. The image of <math>P</math> after these two transformations is at <math>(-6,3)</math>. What is <math>b-a?</math><br />
<br />
<math>\textbf{(A) }1 \qquad \textbf{(B) }3 \qquad \textbf{(C) }5 \qquad \textbf{(D) }7 \qquad \textbf{(E) }9</math><br />
<br />
[[2021 AMC 12B Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
An inverted cone with base radius <math>12 \text{cm}</math> and height <math>18\text{cm}</math> is full of water. The water is poured into a tall cylinder whose horizontal base has a radius of <math>24\text{cm}</math>. What is the height in centimeters of the water in the cylinder?<br />
<br />
<math>\textbf{(A) }1.5 \qquad \textbf{(B) }3 \qquad \textbf{(C) }4 \qquad \textbf{(D) }4.5 \qquad \textbf{(E) }6</math><br />
<br />
[[2021 AMC 12B Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
Let <math>N=34\cdot34\cdot63\cdot270.</math> What is the ratio of the sum of the odd divisors of <math>N</math> to the sum of the even divisors of <math>N?</math><br />
<br />
<math>\textbf{(A) }1:16 \qquad \textbf{(B) }1:15 \qquad \textbf{(C) }1:14 \qquad \textbf{(D) }1:8 \qquad \textbf{(E) }1:3</math><br />
<br />
[[2021 AMC 12B Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
Three equally spaced parallel lines intersect a circle, creating three chords of lengths <math>38,38,</math> and <math>34</math>. What is the distance between two adjacent parallel lines?<br />
<br />
<math>\textbf{(A) }5\frac12 \qquad \textbf{(B) }6 \qquad \textbf{(C) }6\frac12 \qquad \textbf{(D) }7 \qquad \textbf{(E) }7\frac12</math><br />
<br />
[[2021 AMC 12B Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
What is the value of<cmath>\frac{\log_2 80}{\log_{40}2}-\frac{\log_2 160}{\log_{20}2}?</cmath><br />
<math>\textbf{(A) }0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }\frac54 \qquad \textbf{(D) }2 \qquad \textbf{(E) }\log_2 5</math><br />
<br />
[[2021 AMC 12B Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
Two distinct numbers are selected from the set <math>\{1,2,3,4,\dots,36,37\}</math> so that the sum of the remaining <math>35</math> numbers is the product of these two numbers. What is the difference of these two numbers?<br />
<br />
<math>\textbf{(A) }5 \qquad \textbf{(B) }7 \qquad \textbf{(C) }8\qquad \textbf{(D) }9 \qquad \textbf{(E) }10</math><br />
<br />
[[2021 AMC 12B Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
Triangle <math>ABC</math> has <math>AB=13,BC=14</math> and <math>AC=15</math>. Let <math>P</math> be the point on <math>\overline{AC}</math> such that <math>PC=10</math>. There are exactly two points <math>D</math> and <math>E</math> on line <math>BP</math> such that quadrilaterals <math>ABCD</math> and <math>ABCE</math> are trapezoids. What is the distance <math>DE?</math><br />
<br />
<math>\textbf{(A) }\frac{42}5 \qquad \textbf{(B) }6\sqrt2 \qquad \textbf{(C) }\frac{84}5\qquad \textbf{(D) }12\sqrt2 \qquad \textbf{(E) }18</math><br />
<br />
[[2021 AMC 12B Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
Suppose that <math>S</math> is a finite set of positive integers. If the greatest integer in <math>S</math> is removed from <math>S</math>, then the average value (arithmetic mean) of the integers remaining is <math>32</math>. If the least integer in <math>S</math> is also removed, then the average value of the integers remaining is <math>35</math>. If the great integer is then returned to the set, the average value of the integers rises to <math>40.</math> The greatest integer in the original set <math>S</math> is <math>72</math> greater than the least integer in <math>S</math>. What is the average value of all the integers in the set <math>S?</math><br />
<br />
<math>\textbf{(A) }36.2 \qquad \textbf{(B) }36.4 \qquad \textbf{(C) }36.6\qquad \textbf{(D) }36.8 \qquad \textbf{(E) }37</math><br />
<br />
[[2021 AMC 12B Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
How many values of <math>\theta</math> in the interval <math>0<\theta\le 2\pi</math> satisfy<cmath>1-3\sin\theta+5\cos3\theta?</cmath><br />
<math>\textbf{(A) }2 \qquad \textbf{(B) }4 \qquad \textbf{(C) }5\qquad \textbf{(D) }6 \qquad \textbf{(E) }8</math><br />
<br />
[[2021 AMC 12B Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
Let <math>ABCD</math> be a rectangle and let <math>\overline{DM}</math> be a segment perpendicular to the plane of <math>ABCD</math>. Suppose that <math>\overline{DM}</math> has integer length, and the lengths of <math>\overline{MA},\overline{MC},</math> and <math>\overline{MB}</math> are consecutive odd positive integers (in this order). What is the volume of pyramid <math>MACD?</math><br />
<br />
<math>\textbf{(A) }24\sqrt5 \qquad \textbf{(B) }60 \qquad \textbf{(C) }28\sqrt5\qquad \textbf{(D) }66 \qquad \textbf{(E) }8\sqrt{70}</math><br />
<br />
[[2021 AMC 12B Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
The figure is constructed from <math>11</math> line segments, each of which has length <math>2</math>. The area of pentagon <math>ABCDE</math> can be written is <math>\sqrt{m} + \sqrt{n}</math>, where <math>m</math> and <math>n</math> are positive integers. What is <math>m + n ?</math><br />
<asy> /* Made by samrocksnature */ pair A=(-2.4638,4.10658); pair B=(-4,2.6567453480756127); pair C=(-3.47132,0.6335248637894945); pair D=(-1.464483379039766,0.6335248637894945); pair E=(-0.956630463955801,2.6567453480756127); pair F=(-2,2); pair G=(-3,2); draw(A--B--C--D--E--A); draw(A--F--A--G); draw(B--F--C); draw(E--G--D); label("A",A,N); label("B",B,W); label("C",C,W); label("D",D,E); label("E",E,dir(0)); dot(A^^B^^C^^D^^E^^F^^G); </asy><br />
<math>\textbf{(A)} ~20 \qquad\textbf{(B)} ~21 \qquad\textbf{(C)} ~22 \qquad\textbf{(D)} ~23 \qquad\textbf{(E)} ~24</math><br />
<br />
[[2021 AMC 12B Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
Let <math>g(x)</math> be a polynomial with leading coefficient <math>1,</math> whose three roots are the reciprocals of the three roots of <math>f(x)=x^3+ax^2+bx+c,</math> where <math>1<a<b<c.</math> What is <math>g(1)</math> in terms of <math>a,b,</math> and <math>c?</math><br />
<br />
<math>\textbf{(A) }\frac{1+a+b+c}c \qquad \textbf{(B) }1+a+b+c \qquad \textbf{(C) }\frac{1+a+b+c}{c^2}\qquad \textbf{(D) }\frac{a+b+c}{c^2} \qquad \textbf{(E) }\frac{1+a+b+c}{a+b+c}</math><br />
<br />
[[2021 AMC 12B Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
Let <math>ABCD</math> be an isoceles trapezoid having parallel bases <math>\overline{AB}</math> and <math>\overline{CD}</math> with <math>AB>CD.</math> Line segments from a point inside <math>ABCD</math> to the vertices divide the trapezoid into four triangles whose areas are <math>2, 3, 4,</math> and <math>5</math> starting with the triangle with base <math>\overline{CD}</math> and moving clockwise as shown in the diagram below. What is the ratio <math>\frac{AB}{CD}?</math><br />
<asy><br />
unitsize(100);<br />
pair A=(-1, 0), B=(1, 0), C=(0.3, 0.9), D=(-0.3, 0.9), P=(0.2, 0.5), E=(0.1, 0.75), F=(0.4, 0.5), G=(0.15, 0.2), H=(-0.3, 0.5); <br />
draw(A--B--C--D--cycle, black); <br />
draw(A--P, black);<br />
draw(B--P, black);<br />
draw(C--P, black);<br />
draw(D--P, black);<br />
label("$A$",A,(-1,0));<br />
label("$B$",B,(1,0));<br />
label("$C$",C,(1,-0));<br />
label("$D$",D,(-1,0));<br />
label("$2$",E,(0,0));<br />
label("$3$",F,(0,0));<br />
label("$4$",G,(0,0));<br />
label("$5$",H,(0,0));<br />
dot(A^^B^^C^^D^^P);<br />
</asy><br />
<math>\textbf{(A)}\: 3\qquad\textbf{(B)}\: 2+\sqrt{2}\qquad\textbf{(C)}\: 1+\sqrt{6}\qquad\textbf{(D)}\: 2\sqrt{3}\qquad\textbf{(E)}\: 3\sqrt{2}</math><br />
<br />
[[2021 AMC 12B Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
Let <math>z</math> be a complex number satisfying <math>12|z|^2=2|z+2|^2+|z^2+1|^2+31.</math> What is the value of <math>z+\frac 6z?</math><br />
<br />
<math>\textbf{(A) }-2 \qquad \textbf{(B) }-1 \qquad \textbf{(C) }\frac12\qquad \textbf{(D) }1 \qquad \textbf{(E) }4</math><br />
<br />
[[2021 AMC 12B Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
Two fair dice, each with at least <math>6</math> faces are rolled. On each face of each dice is printed a distinct integer from <math>1</math> to the number of faces on that die, inclusive. The probability of rolling a sum if <math>7</math> is <math>\frac34</math> of the probability of rolling a sum of <math>10,</math> and the probability of rolling a sum of <math>12</math> is <math>\frac{1}{12}</math>. What is the least possible number of faces on the two dice combined?<br />
<br />
<math>\textbf{(A) }16 \qquad \textbf{(B) }17 \qquad \textbf{(C) }18\qquad \textbf{(D) }19 \qquad \textbf{(E) }20</math><br />
<br />
[[2021 AMC 12B Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
Let <math>Q(z)</math> and <math>R(z)</math> be the unique polynomials such that<cmath>z^{2021}+1=(z^2+z+1)Q(z)+R(z)</cmath>and the degree of <math>R</math> is less than <math>2.</math> What is <math>R(z)?</math><br />
<br />
<math>\textbf{(A) }-z \qquad \textbf{(B) }-1 \qquad \textbf{(C) }2021\qquad \textbf{(D) }z+1 \qquad \textbf{(E) }2z+1</math><br />
<br />
[[2021 AMC 12B Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
Let <math>S</math> be the sum of all positive real numbers <math>x</math> for which<cmath>x^{2^{\sqrt2}}=\sqrt2^{2^x}.</cmath>Which of the following statements is true?<br />
<br />
<math>\textbf{(A) }S<\sqrt2 \qquad \textbf{(B) }S=\sqrt2 \qquad \textbf{(C) }\sqrt2<S<2\qquad \textbf{(D) }2\le S<6 \qquad \textbf{(E) }S\ge 6</math><br />
<br />
[[2021 AMC 12B Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
Arjun and Beth play a game in which they take turns removing one brick or two adjacent bricks from one "wall" among a set of several walls of bricks, with gaps possibly creating new walls. The walls are one brick tall. For example, a set of walls of sizes <math>4</math> and <math>2</math> can be changed into any of the following by one move: <math>(3,2),(2,1,2),(4),(4,1),(2,2),</math> or <math>(1,1,2).</math> <br />
<br />
<asy><br />
unitsize(4mm);<br />
real[] boxes = {0,1,2,3,5,6,13,14,15,17,18,21,22,24,26,27,30,31,32,33};<br />
for(real i:boxes){<br />
draw(box((i,0),(i+1,3)));<br />
}<br />
draw((8,1.5)--(12,1.5),Arrow());<br />
defaultpen(fontsize(20pt));<br />
label(",",(20,0));<br />
label(",",(29,0));<br />
label(",...",(35.5,0));<br />
</asy><br />
<br />
Arjun plays first, and the player who removes the last brick wins. For which starting configuration is there a strategy that guarantees a win for Beth?<br />
<br />
<math>\textbf{(A) }(6,1,1) \qquad \textbf{(B) }(6,2,1) \qquad \textbf{(C) }(6,2,2)\qquad \textbf{(D) }(6,3,1) \qquad \textbf{(E) }(6,3,2)</math><br />
<br />
[[2021 AMC 12B Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
Three balls are randomly and independantly tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin <math>i</math> is <math>2^{-i}</math> for <math>i=1,2,3,....</math> More than one ball is allowed in each bin. The probability that the balls end up evenly spaced in distinct bins is <math>\frac pq,</math> where <math>p</math> and <math>q</math> are relatively prime positive integers. (For example, the balls are evenly spaced if they are tossed into bins <math>3,17,</math> and <math>10.</math>) What is <math>p+q?</math><br />
<br />
<math>\textbf{(A) }55 \qquad \textbf{(B) }56 \qquad \textbf{(C) }57\qquad \textbf{(D) }58 \qquad \textbf{(E) }59</math><br />
<br />
[[2021 AMC 12B Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
Let <math>ABCD</math> be a parallelogram with area <math>15</math>. Points <math>P</math> and <math>Q</math> are the projections of <math>A</math> and <math>C,</math> respectively, onto the line <math>BD;</math> and points <math>R</math> and <math>S</math> are the projections of <math>B</math> and <math>D,</math> respectively, onto the line <math>AC.</math> See the figure, which also shows the relative locations of these points.<br />
<br />
<asy><br />
size(350);<br />
defaultpen(linewidth(0.8)+fontsize(11));<br />
real theta = aTan(1.25/2);<br />
pair A = 2.5*dir(180+theta), B = (3.35,0), C = -A, D = -B, P = foot(A,B,D), Q = -P, R = foot(B,A,C), S = -R;<br />
draw(A--B--C--D--A^^B--D^^R--S^^rightanglemark(A,P,D,6)^^rightanglemark(C,Q,D,6));<br />
draw(B--R^^C--Q^^A--P^^D--S,linetype("4 4"));<br />
dot("$A$",A,dir(270));<br />
dot("$B$",B,E);<br />
dot("$C$",C,N);<br />
dot("$D$",D,W);<br />
dot("$P$",P,SE);<br />
dot("$Q$",Q,NE);<br />
dot("$R$",R,N);<br />
dot("$S$",S,dir(270));<br />
</asy><br />
<br />
Suppose <math>PQ=6</math> and <math>RS=8,</math> and let <math>d</math> denote the length of <math>\overline{BD},</math> the longer diagonal of <math>ABCD.</math> Then <math>d^2</math> can be written in the form <math>m+n\sqrt p,</math> where <math>m,n,</math> and <math>p</math> are positive integers and <math>p</math> is not divisible by the square of any prime. What is <math>m+n+p?</math><br />
<br />
<math>\textbf{(A) }81 \qquad \textbf{(B) }89 \qquad \textbf{(C) }97\qquad \textbf{(D) }105 \qquad \textbf{(E) }113</math><br />
<br />
[[2021 AMC 12B Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
Let <math>S</math> be the set of lattice points in the coordinate plane, both of whose coordinates are integers between <math>1</math> and <math>30,</math> inclusive. Exactly <math>300</math> points in <math>S</math> lie on or below a line with equation <math>y=mx.</math> The possible values of <math>m</math> lie in an interval of length <math>\frac ab,</math> where <math>a</math> and <math>b</math> are relatively prime positive integers. What is <math>a+b?</math><br />
<br />
<math>\textbf{(A) }31 \qquad \textbf{(B) }47 \qquad \textbf{(C) }62\qquad \textbf{(D) }72 \qquad \textbf{(E) }85</math><br />
<br />
[[2021 AMC 12B Problems/Problem 25|Solution]]<br />
<br />
==See also==<br />
{{AMC12 box|year=2021|ab=B|before=[[2021 AMC 12A Problems]]|after=[[2022 AMC 12A Problems]]}}<br />
<br />
[[Category:AMC 12 Problems]]<br />
{{MAA Notice}}</div>Theajlhttps://artofproblemsolving.com/wiki/index.php?title=2021_AMC_12B_Problems&diff=1457462021 AMC 12B Problems2021-02-12T04:26:16Z<p>Theajl: /* Problem 25 */</p>
<hr />
<div>{{AMC12 Problems|year=2021|ab=B}}<br />
==Problem 1==<br />
How many integer values of <math>x</math> satisfy <math>|x|<3\pi?</math><br />
<br />
<math>\textbf{(A) }9 \qquad \textbf{(B) }10 \qquad \textbf{(C) }18 \qquad \textbf{(D) }19 \qquad \textbf{(E) }20</math><br />
<br />
[[2021 AMC 12B Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
At a math contest, <math>57</math> students are wearing blue shirts, and another <math>75</math> students are wearing yellow shirts. The <math>132</math> students are assigned into <math>66</math> points. In exactly <math>23</math> of these pairs, both students are wearing blue shirts. In how many pairs are both studets wearing yellow shirts?<br />
<br />
<math>\textbf{(A) }23 \qquad \textbf{(B) }32 \qquad \textbf{(C) }37 \qquad \textbf{(D) }41 \qquad \textbf{(E) }64</math><br />
<br />
[[2021 AMC 12B Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
Suppose<cmath>2+\frac{1}{1+\frac{1}{2+\frac{2}{3+x}}}=\frac{144}{53}.</cmath>What is the value of <math>x?</math><br />
<br />
<math>\textbf{(A) }\frac34 \qquad \textbf{(B) }\frac78 \qquad \textbf{(C) }\frac{14}{15} \qquad \textbf{(D) }\frac{37}{38} \qquad \textbf{(E) }\frac{52}{53}</math><br />
<br />
[[2021 AMC 12B Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
Ms. Blackwell gives an exam to two classes. The mean of the scores of the students in the morning class is <math>84</math>, and the afternoon class's mean score is <math>70</math>. The ratio of the number of students in the morning clas to the number of students in the afternoon class is <math>\frac34</math>. What is the mean of the score of all the students?<br />
<br />
<math>\textbf{(A) }74 \qquad \textbf{(B) }75 \qquad \textbf{(C) }76 \qquad \textbf{(D) }77 \qquad \textbf{(E) }78</math><br />
<br />
[[2021 AMC 12B Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
The point <math>P(a,b)</math> in the <math>xy</math>-plane is first rotated counterclockwise by <math>90^\circ</math> around the point <math>(1,5)</math> and then reflected about the line <math>y=-x</math>. The image of <math>P</math> after these two transformations is at <math>(-6,3)</math>. What is <math>b-a?</math><br />
<br />
<math>\textbf{(A) }1 \qquad \textbf{(B) }3 \qquad \textbf{(C) }5 \qquad \textbf{(D) }7 \qquad \textbf{(E) }9</math><br />
<br />
[[2021 AMC 12B Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
An inverted cone with base radius <math>12 \text{cm}</math> and height <math>18\text{cm}</math> is full of water. The water is poured into a tall cylinder whose horizontal base has a radius of <math>24\text{cm}</math>. What is the height in centimeters of the water in the cylinder?<br />
<br />
<math>\textbf{(A) }1.5 \qquad \textbf{(B) }3 \qquad \textbf{(C) }4 \qquad \textbf{(D) }4.5 \qquad \textbf{(E) }6</math><br />
<br />
[[2021 AMC 12B Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
Let <math>N=34\cdot34\cdot63\cdot270.</math> What is the ratio of the sum of the odd divisors of <math>N</math> to the sum of the even divisors of <math>N?</math><br />
<br />
<math>\textbf{(A) }1:16 \qquad \textbf{(B) }1:15 \qquad \textbf{(C) }1:14 \qquad \textbf{(D) }1:8 \qquad \textbf{(E) }1:3</math><br />
<br />
[[2021 AMC 12B Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
Three equally spaced parallel lines intersect a circle, creating three chords of lengths <math>38,38,</math> and <math>34</math>. What is the distance between two adjacent parallel lines?<br />
<br />
<math>\textbf{(A) }5\frac12 \qquad \textbf{(B) }6 \qquad \textbf{(C) }6\frac12 \qquad \textbf{(D) }7 \qquad \textbf{(E) }7\frac12</math><br />
<br />
[[2021 AMC 12B Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
What is the value of<cmath>\frac{\log_2 80}{\log_{40}2}-\frac{\log_2 160}{\log_{20}2}?</cmath><br />
<math>\textbf{(A) }0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }\frac54 \qquad \textbf{(D) }2 \qquad \textbf{(E) }\log_2 5</math><br />
<br />
[[2021 AMC 12B Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
Two distinct numbers are selected from the set <math>\{1,2,3,4,\dots,36,37\}</math> so that the sum of the remaining <math>35</math> numbers is the product of these two numbers. What is the difference of these two numbers?<br />
<br />
<math>\textbf{(A) }5 \qquad \textbf{(B) }7 \qquad \textbf{(C) }8\qquad \textbf{(D) }9 \qquad \textbf{(E) }10</math><br />
<br />
[[2021 AMC 12B Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
Triangle <math>ABC</math> has <math>AB=13,BC=14</math> and <math>AC=15</math>. Let <math>P</math> be the point on <math>\overline{AC}</math> such that <math>PC=10</math>. There are exactly two points <math>D</math> and <math>E</math> on line <math>BP</math> such that quadrilaterals <math>ABCD</math> and <math>ABCE</math> are trapezoids. What is the distance <math>DE?</math><br />
<br />
<math>\textbf{(A) }\frac{42}5 \qquad \textbf{(B) }6\sqrt2 \qquad \textbf{(C) }\frac{84}5\qquad \textbf{(D) }12\sqrt2 \qquad \textbf{(E) }18</math><br />
<br />
[[2021 AMC 12B Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
Suppose that <math>S</math> is a finite set of positive integers. If the greatest integer in <math>S</math> is removed from <math>S</math>, then the average value (arithmetic mean) of the integers remaining is <math>32</math>. If the least integer in <math>S</math> is also removed, then the average value of the integers remaining is <math>35</math>. If the great integer is then returned to the set, the average value of the integers rises to <math>40.</math> The greatest integer in the original set <math>S</math> is <math>72</math> greater than the least integer in <math>S</math>. What is the average value of all the integers in the set <math>S?</math><br />
<br />
<math>\textbf{(A) }36.2 \qquad \textbf{(B) }36.4 \qquad \textbf{(C) }36.6\qquad \textbf{(D) }36.8 \qquad \textbf{(E) }37</math><br />
<br />
[[2021 AMC 12B Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
How many values of <math>\theta</math> in the interval <math>0<\theta\le 2\pi</math> satisfy<cmath>1-3\sin\theta+5\cos3\theta?</cmath><br />
<math>\textbf{(A) }2 \qquad \textbf{(B) }4 \qquad \textbf{(C) }5\qquad \textbf{(D) }6 \qquad \textbf{(E) }8</math><br />
<br />
[[2021 AMC 12B Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
Let <math>ABCD</math> be a rectangle and let <math>\overline{DM}</math> be a segment perpendicular to the plane of <math>ABCD</math>. Suppose that <math>\overline{DM}</math> has integer length, and the lengths of <math>\overline{MA},\overline{MC},</math> and <math>\overline{MB}</math> are consecutive odd positive integers (in this order). What is the volume of pyramid <math>MACD?</math><br />
<br />
<math>\textbf{(A) }24\sqrt5 \qquad \textbf{(B) }60 \qquad \textbf{(C) }28\sqrt5\qquad \textbf{(D) }66 \qquad \textbf{(E) }8\sqrt{70}</math><br />
<br />
[[2021 AMC 12B Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
The figure is constructed from <math>11</math> line segments, each of which has length <math>2</math>. The area of pentagon <math>ABCDE</math> can be written is <math>\sqrt{m} + \sqrt{n}</math>, where <math>m</math> and <math>n</math> are positive integers. What is <math>m + n ?</math><br />
<asy> /* Made by samrocksnature */ pair A=(-2.4638,4.10658); pair B=(-4,2.6567453480756127); pair C=(-3.47132,0.6335248637894945); pair D=(-1.464483379039766,0.6335248637894945); pair E=(-0.956630463955801,2.6567453480756127); pair F=(-2,2); pair G=(-3,2); draw(A--B--C--D--E--A); draw(A--F--A--G); draw(B--F--C); draw(E--G--D); label("A",A,N); label("B",B,W); label("C",C,W); label("D",D,E); label("E",E,dir(0)); dot(A^^B^^C^^D^^E^^F^^G); </asy><br />
<math>\textbf{(A)} ~20 \qquad\textbf{(B)} ~21 \qquad\textbf{(C)} ~22 \qquad\textbf{(D)} ~23 \qquad\textbf{(E)} ~24</math><br />
<br />
[[2021 AMC 12B Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
Let <math>g(x)</math> be a polynomial with leading coefficient <math>1,</math> whose three roots are the reciprocals of the three roots of <math>f(x)=x^3+ax^2+bx+c,</math> where <math>1<a<b<c.</math> What is <math>g(1)</math> in terms of <math>a,b,</math> and <math>c?</math><br />
<br />
<math>\textbf{(A) }\frac{1+a+b+c}c \qquad \textbf{(B) }1+a+b+c \qquad \textbf{(C) }\frac{1+a+b+c}{c^2}\qquad \textbf{(D) }\frac{a+b+c}{c^2} \qquad \textbf{(E) }\frac{1+a+b+c}{a+b+c}</math><br />
<br />
[[2021 AMC 12B Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
Let <math>ABCD</math> be an isoceles trapezoid having parallel bases <math>\overline{AB}</math> and <math>\overline{CD}</math> with <math>AB>CD.</math> Line segments from a point inside <math>ABCD</math> to the vertices divide the trapezoid into four triangles whose areas are <math>2, 3, 4,</math> and <math>5</math> starting with the triangle with base <math>\overline{CD}</math> and moving clockwise as shown in the diagram below. What is the ratio <math>\frac{AB}{CD}?</math><br />
<asy><br />
unitsize(100);<br />
pair A=(-1, 0), B=(1, 0), C=(0.3, 0.9), D=(-0.3, 0.9), P=(0.2, 0.5), E=(0.1, 0.75), F=(0.4, 0.5), G=(0.15, 0.2), H=(-0.3, 0.5); <br />
draw(A--B--C--D--cycle, black); <br />
draw(A--P, black);<br />
draw(B--P, black);<br />
draw(C--P, black);<br />
draw(D--P, black);<br />
label("$A$",A,(-1,0));<br />
label("$B$",B,(1,0));<br />
label("$C$",C,(1,-0));<br />
label("$D$",D,(-1,0));<br />
label("$2$",E,(0,0));<br />
label("$3$",F,(0,0));<br />
label("$4$",G,(0,0));<br />
label("$5$",H,(0,0));<br />
dot(A^^B^^C^^D^^P);<br />
</asy><br />
<math>\textbf{(A)}\: 3\qquad\textbf{(B)}\: 2+\sqrt{2}\qquad\textbf{(C)}\: 1+\sqrt{6}\qquad\textbf{(D)}\: 2\sqrt{3}\qquad\textbf{(E)}\: 3\sqrt{2}</math><br />
<br />
[[2021 AMC 12B Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
Let <math>z</math> be a complex number satisfying <math>12|z|^2=2|z+2|^2+|z^2+1|^2+31.</math> What is the value of <math>z+\frac 6z?</math><br />
<br />
<math>\textbf{(A) }-2 \qquad \textbf{(B) }-1 \qquad \textbf{(C) }\frac12\qquad \textbf{(D) }1 \qquad \textbf{(E) }4</math><br />
<br />
[[2021 AMC 12B Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
Two fair dice, each with at least <math>6</math> faces are rolled. On each face of each dice is printed a distinct integer from <math>1</math> to the number of faces on that die, inclusive. The probability of rolling a sum if <math>7</math> is <math>\frac34</math> of the probability of rolling a sum of <math>10,</math> and the probability of rolling a sum of <math>12</math> is <math>\frac{1}{12}</math>. What is the least possible number of faces on the two dice combined?<br />
<br />
<math>\textbf{(A) }16 \qquad \textbf{(B) }17 \qquad \textbf{(C) }18\qquad \textbf{(D) }19 \qquad \textbf{(E) }20</math><br />
<br />
[[2021 AMC 12B Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
Let <math>Q(z)</math> and <math>R(z)</math> be the unique polynomials such that<cmath>z^{2021}+1=(z^2+z+1)Q(z)+R(z)</cmath>and the degree of <math>R</math> is less than <math>2.</math> What is <math>R(z)?</math><br />
<br />
<math>\textbf{(A) }-z \qquad \textbf{(B) }-1 \qquad \textbf{(C) }2021\qquad \textbf{(D) }z+1 \qquad \textbf{(E) }2z+1</math><br />
<br />
[[2021 AMC 12B Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
Let <math>S</math> be the sum of all positive real numbers <math>x</math> for which<cmath>x^{2^{\sqrt2}}=\sqrt2^{2^x}.</cmath>Which of the following statements is true?<br />
<br />
<math>\textbf{(A) }S<\sqrt2 \qquad \textbf{(B) }S=\sqrt2 \qquad \textbf{(C) }\sqrt2<S<2\qquad \textbf{(D) }2\le S<6 \qquad \textbf{(E) }S\ge 6</math><br />
<br />
[[2021 AMC 12B Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
Arjun and Beth play a game in which they take turns removing one brick or two adjacent bricks from one "wall" among a set of several walls of bricks, with gaps possibly creating new walls. The walls are one brick tall. For example, a set of walls of sizes <math>4</math> and <math>2</math> can be changed into any of the following by one move: <math>(3,2),(2,1,2),(4),(4,1),(2,2),</math> or <math>(1,1,2).</math> <br />
<br />
<asy><br />
unitsize(4mm);<br />
real[] boxes = {0,1,2,3,5,6,13,14,15,17,18,21,22,24,26,27,30,31,32,33};<br />
for(real i:boxes){<br />
draw(box((i,0),(i+1,3)));<br />
}<br />
draw((8,1.5)--(12,1.5),Arrow());<br />
defaultpen(fontsize(20pt));<br />
label(",",(20,0));<br />
label(",",(29,0));<br />
label(",...",(35.5,0));<br />
</asy><br />
<br />
Arjun plays first, and the player who removes the last brick wins. For which starting configuration is there a strategy that guarantees a win for Beth?<br />
<br />
<math>\textbf{(A) }(6,1,1) \qquad \textbf{(B) }(6,2,1) \qquad \textbf{(C) }(6,2,2)\qquad \textbf{(D) }(6,3,1) \qquad \textbf{(E) }(6,3,2)</math><br />
<br />
[[2021 AMC 12B Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
Three balls are randomly and independantly tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin <math>i</math> is <math>2^{-i}</math> for <math>i=1,2,3,....</math> More than one ball is allowed in each bin. The probability that the balls end up evenly spaced in distinct bins is <math>\frac pq,</math> where <math>p</math> and <math>q</math> are relatively prime positive integers. (For example, the balls are evenly spaced if they are tossed into bins <math>3,17,</math> and <math>10.</math>) What is <math>p+q?</math><br />
<br />
<math>\textbf{(A) }55 \qquad \textbf{(B) }56 \qquad \textbf{(C) }57\qquad \textbf{(D) }58 \qquad \textbf{(E) }59</math><br />
<br />
[[2021 AMC 12B Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
Let <math>ABCD</math> be a parallelogram with area <math>15</math>. Points <math>P</math> and <math>Q</math> are the projections of <math>A</math> and <math>C,</math> respectively, onto the line <math>BD;</math> and points <math>R</math> and <math>S</math> are the projections of <math>B</math> and <math>D,</math> respectively, onto the line <math>AC.</math> See the figure, which also shows the relative locations of these points.<br />
<br />
<asy><br />
size(350);<br />
defaultpen(linewidth(0.8)+fontsize(11));<br />
real theta = aTan(1.25/2);<br />
pair A = 2.5*dir(180+theta), B = (3.35,0), C = -A, D = -B, P = foot(A,B,D), Q = -P, R = foot(B,A,C), S = -R;<br />
draw(A--B--C--D--A^^B--D^^R--S^^rightanglemark(A,P,D,6)^^rightanglemark(C,Q,D,6));<br />
draw(B--R^^C--Q^^A--P^^D--S,linetype("4 4"));<br />
dot("$A$",A,dir(270));<br />
dot("$B$",B,E);<br />
dot("$C$",C,N);<br />
dot("$D$",D,W);<br />
dot("$P$",P,SE);<br />
dot("$Q$",Q,NE);<br />
dot("$R$",R,N);<br />
dot("$S$",S,dir(270));<br />
</asy><br />
<br />
Suppose <math>PQ=6</math> and <math>RS=8,</math> and let <math>d</math> denote the length of <math>\overline{BD},</math> the longer diagonal of <math>ABCD.</math> Then <math>d^2</math> can be written in the form <math>m+n\sqrt p,</math> where <math>m,n,</math> and <math>p</math> are positive integers and <math>p</math> is not divisible by the square of any prime. What is <math>m+n+p?</math><br />
<br />
<math>\textbf{(A) }81 \qquad \textbf{(B) }89 \qquad \textbf{(C) }97\qquad \textbf{(D) }105 \qquad \textbf{(E) }113</math><br />
<br />
[[2021 AMC 12B Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
Let <math>S</math> be the set of lattice points in the coordinate plane, both of whose coordinates are integers between <math>1</math> and <math>30,</math> inclusive. Exactly <math>300</math> points in <math>S</math> lie on or below a line with equation <math>y=mx.</math> The possible values of <math>m</math> lie in an interval of length <math>\frac ab,</math> where <math>a</math> and <math>b</math> are relatively prime positive integers. What is <math>a+b?</math><br />
<br />
<math>\textbf{(A) }31 \qquad \textbf{(B) }47 \qquad \textbf{(C) }62\qquad \textbf{(D) }72 \qquad \textbf{(E) }85</math><br />
<br />
[[2021 AMC 12B Problems/Problem 25|Solution]]<br />
<br />
<br />
<br />
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 900 lattice points. I estimate that the slope, m, 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 600 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 are is above the line. <br />
<br />
I find that the number of coordinates with <math>x=1</math> above the line is 30, 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 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 <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 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. <br />
<br />
To find the upper bound, notice that each point with an integer-valued x-coordinate is either <math>\frac{1}{3}</math> or <math>\frac{2}{3}</math> 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 <math>x=28, 29, 30</math> which the line <math>y=\frac{2}{3}x</math> intersects at <math>y= \frac{56}{3}, \frac{58}{3}, 20</math>. 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, <math>\frac{56}{3}</math>) is closer to the lattice point. The slope of the line which goes through both the origin and (28,19) is <math>y=\frac{19}{28}x</math>. This gives an upper bound of <math>m=\frac{19}{28}</math>. <br />
<br />
Taking the upper bound of m and subtracting the lower bound yields <math>\frac{19}{28}-\frac{2}{3}=\frac{1}{84}</math>. This is answer E.<br />
<br />
<br />
~Theajl<br />
<br />
==See also==<br />
{{AMC12 box|year=2021|ab=B|before=[[2021 AMC 12A Problems]]|after=[[2022 AMC 12A Problems]]}}<br />
<br />
[[Category:AMC 12 Problems]]<br />
{{MAA Notice}}</div>Theajlhttps://artofproblemsolving.com/wiki/index.php?title=2021_AMC_12B_Problems&diff=1457282021 AMC 12B Problems2021-02-12T04:03:48Z<p>Theajl: /* Problem 25 */</p>
<hr />
<div>{{AMC12 Problems|year=2021|ab=B}}<br />
==Problem 1==<br />
How many integer values of <math>x</math> satisfy <math>|x|<3\pi?</math><br />
<br />
<math>\textbf{(A) }9 \qquad \textbf{(B) }10 \qquad \textbf{(C) }18 \qquad \textbf{(D) }19 \qquad \textbf{(E) }20</math><br />
<br />
[[2021 AMC 12B Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
At a math contest, <math>57</math> students are wearing blue shirts, and another <math>75</math> students are wearing yellow shirts. The <math>132</math> students are assigned into <math>66</math> points. In exactly <math>23</math> of these pairs, both students are wearing blue shirts. In how many pairs are both studets wearing yellow shirts?<br />
<br />
<math>\textbf{(A) }23 \qquad \textbf{(B) }32 \qquad \textbf{(C) }37 \qquad \textbf{(D) }41 \qquad \textbf{(E) }64</math><br />
<br />
[[2021 AMC 12B Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
Suppose<cmath>2+\frac{1}{1+\frac{1}{2+\frac{2}{3+x}}}=\frac{144}{53}.</cmath>What is the value of <math>x?</math><br />
<br />
<math>\textbf{(A) }\frac34 \qquad \textbf{(B) }\frac78 \qquad \textbf{(C) }\frac{14}{15} \qquad \textbf{(D) }\frac{37}{38} \qquad \textbf{(E) }\frac{52}{53}</math><br />
<br />
[[2021 AMC 12B Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
Ms. Blackwell gives an exam to two classes. The mean of the scores of the students in the morning class is <math>84</math>, and the afternoon class's mean score is <math>70</math>. The ratio of the number of students in the morning clas to the number of students in the afternoon class is <math>\frac34</math>. What is the mean of the score of all the students?<br />
<br />
<math>\textbf{(A) }74 \qquad \textbf{(B) }75 \qquad \textbf{(C) }76 \qquad \textbf{(D) }77 \qquad \textbf{(E) }78</math><br />
<br />
[[2021 AMC 12B Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
The point <math>P(a,b)</math> in the <math>xy</math>-plane is first rotated counterclockwise by <math>90^\circ</math> around the point <math>(1,5)</math> and then reflected about the line <math>y=-x</math>. The image of <math>P</math> after these two transformations is at <math>(-6,3)</math>. What is <math>b-a?</math><br />
<br />
<math>\textbf{(A) }1 \qquad \textbf{(B) }3 \qquad \textbf{(C) }5 \qquad \textbf{(D) }7 \qquad \textbf{(E) }9</math><br />
<br />
[[2021 AMC 12B Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
An inverted cone with base radius <math>12 \text{cm}</math> and height <math>18\text{cm}</math> is full of water. The water is poured into a tall cylinder whose horizontal base has a radius of <math>24\text{cm}</math>. What is the height in centimeters of the water in the cylinder?<br />
<br />
<math>\textbf{(A) }1.5 \qquad \textbf{(B) }3 \qquad \textbf{(C) }4 \qquad \textbf{(D) }4.5 \qquad \textbf{(E) }6</math><br />
<br />
[[2021 AMC 12B Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
Let <math>N=34\cdot34\cdot63\cdot270.</math> What is the ratio of the sum of the odd divisors of <math>N</math> to the sum of the even divisors of <math>N?</math><br />
<br />
<math>\textbf{(A) }1:16 \qquad \textbf{(B) }1:15 \qquad \textbf{(C) }1:14 \qquad \textbf{(D) }1:8 \qquad \textbf{(E) }1:3</math><br />
<br />
[[2021 AMC 12B Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
Three equally spaced parallel lines intersect a circle, creating three chords of lengths <math>38,38,</math> and <math>34</math>. What is the distance between two adjacent parallel lines?<br />
<br />
<math>\textbf{(A) }5\frac12 \qquad \textbf{(B) }6 \qquad \textbf{(C) }6\frac12 \qquad \textbf{(D) }7 \qquad \textbf{(E) }7\frac12</math><br />
<br />
[[2021 AMC 12B Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
What is the value of<cmath>\frac{\log_2 80}{\log_{40}2}-\frac{\log_2 160}{\log_{20}2}?</cmath><br />
<math>\textbf{(A) }0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }\frac54 \qquad \textbf{(D) }2 \qquad \textbf{(E) }\log_2 5</math><br />
<br />
[[2021 AMC 12B Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
Two distinct numbers are selected from the set <math>\{1,2,3,4,\dots,36,37\}</math> so that the sum of the remaining <math>35</math> numbers is the product of these two numbers. What is the difference of these two numbers?<br />
<br />
<math>\textbf{(A) }5 \qquad \textbf{(B) }7 \qquad \textbf{(C) }8\qquad \textbf{(D) }9 \qquad \textbf{(E) }10</math><br />
<br />
[[2021 AMC 12B Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
Triangle <math>ABC</math> has <math>AB=13,BC=14</math> and <math>AC=15</math>. Let <math>P</math> be the point on <math>\overline{AC}</math> such that <math>PC=10</math>. There are exactly two points <math>D</math> and <math>E</math> on line <math>BP</math> such that quadrilaterals <math>ABCD</math> and <math>ABCE</math> are trapezoids. What is the distance <math>DE?</math><br />
<br />
<math>\textbf{(A) }\frac{42}5 \qquad \textbf{(B) }6\sqrt2 \qquad \textbf{(C) }\frac{84}5\qquad \textbf{(D) }12\sqrt2 \qquad \textbf{(E) }18</math><br />
<br />
[[2021 AMC 12B Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
Suppose that <math>S</math> is a finite set of positive integers. If the greatest integer in <math>S</math> is removed from <math>S</math>, then the average value (arithmetic mean) of the integers remaining is <math>32</math>. If the least integer in <math>S</math> is also removed, then the average value of the integers remaining is <math>35</math>. If the great integer is then returned to the set, the average value of the integers rises to <math>40.</math> The greatest integer in the original set <math>S</math> is <math>72</math> greater than the least integer in <math>S</math>. What is the average value of all the integers in the set <math>S?</math><br />
<br />
<math>\textbf{(A) }36.2 \qquad \textbf{(B) }36.4 \qquad \textbf{(C) }36.6\qquad \textbf{(D) }36.8 \qquad \textbf{(E) }37</math><br />
<br />
[[2021 AMC 12B Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
How many values of <math>\theta</math> in the interval <math>0<\theta\le 2\pi</math> satisfy<cmath>1-3\sin\theta+5\cos3\theta?</cmath><br />
<math>\textbf{(A) }2 \qquad \textbf{(B) }4 \qquad \textbf{(C) }5\qquad \textbf{(D) }6 \qquad \textbf{(E) }8</math><br />
<br />
[[2021 AMC 12B Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
Let <math>ABCD</math> be a rectangle and let <math>\overline{DM}</math> be a segment perpendicular to the plane of <math>ABCD</math>. Suppose that <math>\overline{DM}</math> has integer length, and the lengths of <math>\overline{MA},\overline{MC},</math> and <math>\overline{MB}</math> are consecutive odd positive integers (in this order). What is the volume of pyramid <math>MACD?</math><br />
<br />
<math>\textbf{(A) }24\sqrt5 \qquad \textbf{(B) }60 \qquad \textbf{(C) }28\sqrt5\qquad \textbf{(D) }66 \qquad \textbf{(E) }8\sqrt{70}</math><br />
<br />
[[2021 AMC 12B Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
The figure is constructed from <math>11</math> line segments, each of which has length <math>2</math>. The area of pentagon <math>ABCDE</math> can be written is <math>\sqrt{m} + \sqrt{n}</math>, where <math>m</math> and <math>n</math> are positive integers. What is <math>m + n ?</math><br />
<asy> /* Made by samrocksnature */ pair A=(-2.4638,4.10658); pair B=(-4,2.6567453480756127); pair C=(-3.47132,0.6335248637894945); pair D=(-1.464483379039766,0.6335248637894945); pair E=(-0.956630463955801,2.6567453480756127); pair F=(-2,2); pair G=(-3,2); draw(A--B--C--D--E--A); draw(A--F--A--G); draw(B--F--C); draw(E--G--D); label("A",A,N); label("B",B,W); label("C",C,W); label("D",D,E); label("E",E,dir(0)); dot(A^^B^^C^^D^^E^^F^^G); </asy><br />
<math>\textbf{(A)} ~20 \qquad\textbf{(B)} ~21 \qquad\textbf{(C)} ~22 \qquad\textbf{(D)} ~23 \qquad\textbf{(E)} ~24</math><br />
<br />
[[2021 AMC 12B Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
Let <math>g(x)</math> be a polynomial with leading coefficient <math>1,</math> whose three roots are the reciprocals of the three roots of <math>f(x)=x^3+ax^2+bx+c,</math> where <math>1<a<b<c.</math> What is <math>g(1)</math> in terms of <math>a,b,</math> and <math>c?</math><br />
<br />
<math>\textbf{(A) }\frac{1+a+b+c}c \qquad \textbf{(B) }1+a+b+c \qquad \textbf{(C) }\frac{1+a+b+c}{c^2}\qquad \textbf{(D) }\frac{a+b+c}{c^2} \qquad \textbf{(E) }\frac{1+a+b+c}{a+b+c}</math><br />
<br />
[[2021 AMC 12B Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
Let <math>ABCD</math> be an isoceles trapezoid having parallel bases <math>\overline{AB}</math> and <math>\overline{CD}</math> with <math>AB>CD.</math> Line segments from a point inside <math>ABCD</math> to the vertices divide the trapezoid into four triangles whose areas are <math>2, 3, 4,</math> and <math>5</math> starting with the triangle with base <math>\overline{CD}</math> and moving clockwise as shown in the diagram below. What is the ratio <math>\frac{AB}{CD}?</math><br />
<asy><br />
unitsize(100);<br />
pair A=(-1, 0), B=(1, 0), C=(0.3, 0.9), D=(-0.3, 0.9), P=(0.2, 0.5), E=(0.1, 0.75), F=(0.4, 0.5), G=(0.15, 0.2), H=(-0.3, 0.5); <br />
draw(A--B--C--D--cycle, black); <br />
draw(A--P, black);<br />
draw(B--P, black);<br />
draw(C--P, black);<br />
draw(D--P, black);<br />
label("$A$",A,(-1,0));<br />
label("$B$",B,(1,0));<br />
label("$C$",C,(1,-0));<br />
label("$D$",D,(-1,0));<br />
label("$2$",E,(0,0));<br />
label("$3$",F,(0,0));<br />
label("$4$",G,(0,0));<br />
label("$5$",H,(0,0));<br />
dot(A^^B^^C^^D^^P);<br />
</asy><br />
<math>\textbf{(A)}\: 3\qquad\textbf{(B)}\: 2+\sqrt{2}\qquad\textbf{(C)}\: 1+\sqrt{6}\qquad\textbf{(D)}\: 2\sqrt{3}\qquad\textbf{(E)}\: 3\sqrt{2}</math><br />
<br />
[[2021 AMC 12B Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
Let <math>z</math> be a complex number satisfying <math>12|z|^2=2|z+2|^2+|z^2+1|^2+31.</math> What is the value of <math>z+\frac 6z?</math><br />
<br />
<math>\textbf{(A) }-2 \qquad \textbf{(B) }-1 \qquad \textbf{(C) }\frac12\qquad \textbf{(D) }1 \qquad \textbf{(E) }4</math><br />
<br />
[[2021 AMC 12B Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
Two fair dice, each with at least <math>6</math> faces are rolled. On each face of each dice is printed a distinct integer from <math>1</math> to the number of faces on that die, inclusive. The probability of rolling a sum if <math>7</math> is <math>\frac34</math> of the probability of rolling a sum of <math>10,</math> and the probability of rolling a sum of <math>12</math> is <math>\frac{1}{12}</math>. What is the least possible number of faces on the two dice combined?<br />
<br />
<math>\textbf{(A) }16 \qquad \textbf{(B) }17 \qquad \textbf{(C) }18\qquad \textbf{(D) }19 \qquad \textbf{(E) }20</math><br />
<br />
[[2021 AMC 12B Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
Let <math>Q(z)</math> and <math>R(z)</math> be the unique polynomials such that<cmath>z^{2021}+1=(z^2+z+1)Q(z)+R(z)</cmath>and the degree of <math>R</math> is less than <math>2.</math> What is <math>R(z)?</math><br />
<br />
<math>\textbf{(A) }-z \qquad \textbf{(B) }-1 \qquad \textbf{(C) }2021\qquad \textbf{(D) }z+1 \qquad \textbf{(E) }2z+1</math><br />
<br />
[[2021 AMC 12B Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
Let <math>S</math> be the sum of all positive real numbers <math>x</math> for which<cmath>x^{2^{\sqrt2}}=\sqrt2^{2^x}.</cmath>Which of the following statements is true?<br />
<br />
<math>\textbf{(A) }S<\sqrt2 \qquad \textbf{(B) }S=\sqrt2 \qquad \textbf{(C) }\sqrt2<S<2\qquad \textbf{(D) }2\le S<6 \qquad \textbf{(E) }S\ge 6</math><br />
<br />
[[2021 AMC 12B Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
Arjun and Beth play a game in which they take turns removing one brick or two adjacent bricks from one "wall" among a set of several walls of bricks, with gaps possibly creating new walls. The walls are one brick tall. For example, a set of walls of sizes <math>4</math> and <math>2</math> can be changed into any of the following by one move: <math>(3,2),(2,1,2),(4),(4,1),(2,2),</math> or <math>(1,1,2).</math> <br />
<br />
<asy><br />
unitsize(4mm);<br />
real[] boxes = {0,1,2,3,5,6,13,14,15,17,18,21,22,24,26,27,30,31,32,33};<br />
for(real i:boxes){<br />
draw(box((i,0),(i+1,3)));<br />
}<br />
draw((8,1.5)--(12,1.5),Arrow());<br />
defaultpen(fontsize(20pt));<br />
label(",",(20,0));<br />
label(",",(29,0));<br />
label(",...",(35.5,0));<br />
</asy><br />
<br />
Arjun plays first, and the player who removes the last brick wins. For which starting configuration is there a strategy that guarantees a win for Beth?<br />
<br />
<math>\textbf{(A) }(6,1,1) \qquad \textbf{(B) }(6,2,1) \qquad \textbf{(C) }(6,2,2)\qquad \textbf{(D) }(6,3,1) \qquad \textbf{(E) }(6,3,2)</math><br />
<br />
[[2021 AMC 12B Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
Three balls are randomly and independantly tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin <math>i</math> is <math>2^{-i}</math> for <math>i=1,2,3,....</math> More than one ball is allowed in each bin. The probability that the balls end up evenly spaced in distinct bins is <math>\frac pq,</math> where <math>p</math> and <math>q</math> are relatively prime positive integers. (For example, the balls are evenly spaced if they are tossed into bins <math>3,17,</math> and <math>10.</math>) What is <math>p+q?</math><br />
<br />
<math>\textbf{(A) }55 \qquad \textbf{(B) }56 \qquad \textbf{(C) }57\qquad \textbf{(D) }58 \qquad \textbf{(E) }59</math><br />
<br />
[[2021 AMC 12B Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
Let <math>ABCD</math> be a parallelogram with area <math>15</math>. Points <math>P</math> and <math>Q</math> are the projections of <math>A</math> and <math>C,</math> respectively, onto the line <math>BD;</math> and points <math>R</math> and <math>S</math> are the projections of <math>B</math> and <math>D,</math> respectively, onto the line <math>AC.</math> See the figure, which also shows the relative locations of these points.<br />
<br />
<asy><br />
size(350);<br />
defaultpen(linewidth(0.8)+fontsize(11));<br />
real theta = aTan(1.25/2);<br />
pair A = 2.5*dir(180+theta), B = (3.35,0), C = -A, D = -B, P = foot(A,B,D), Q = -P, R = foot(B,A,C), S = -R;<br />
draw(A--B--C--D--A^^B--D^^R--S^^rightanglemark(A,P,D,6)^^rightanglemark(C,Q,D,6));<br />
draw(B--R^^C--Q^^A--P^^D--S,linetype("4 4"));<br />
dot("$A$",A,dir(270));<br />
dot("$B$",B,E);<br />
dot("$C$",C,N);<br />
dot("$D$",D,W);<br />
dot("$P$",P,SE);<br />
dot("$Q$",Q,NE);<br />
dot("$R$",R,N);<br />
dot("$S$",S,dir(270));<br />
</asy><br />
<br />
Suppose <math>PQ=6</math> and <math>RS=8,</math> and let <math>d</math> denote the length of <math>\overline{BD},</math> the longer diagonal of <math>ABCD.</math> Then <math>d^2</math> can be written in the form <math>m+n\sqrt p,</math> where <math>m,n,</math> and <math>p</math> are positive integers and <math>p</math> is not divisible by the square of any prime. What is <math>m+n+p?</math><br />
<br />
<math>\textbf{(A) }81 \qquad \textbf{(B) }89 \qquad \textbf{(C) }97\qquad \textbf{(D) }105 \qquad \textbf{(E) }113</math><br />
<br />
[[2021 AMC 12B Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
Let <math>S</math> be the set of lattice points in the coordinate plane, both of whose coordinates are integers between <math>1</math> and <math>30,</math> inclusive. Exactly <math>300</math> points in <math>S</math> lie on or below a line with equation <math>y=mx.</math> The possible values of <math>m</math> lie in an interval of length <math>\frac ab,</math> where <math>a</math> and <math>b</math> are relatively prime positive integers. What is <math>a+b?</math><br />
<br />
<math>\textbf{(A) }31 \qquad \textbf{(B) }47 \qquad \textbf{(C) }62\qquad \textbf{(D) }72 \qquad \textbf{(E) }85</math><br />
<br />
[[2021 AMC 12B Problems/Problem 25|Solution]]<br />
<br />
==See also==<br />
{{AMC12 box|year=2021|ab=B|before=[[2021 AMC 12A Problems]]|after=[[2022 AMC 12A Problems]]}}<br />
<br />
[[Category:AMC 12 Problems]]<br />
{{MAA Notice}}</div>Theajlhttps://artofproblemsolving.com/wiki/index.php?title=2021_AMC_12B_Problems&diff=1457202021 AMC 12B Problems2021-02-12T03:59:53Z<p>Theajl: /* Problem 25 */</p>
<hr />
<div>{{AMC12 Problems|year=2021|ab=B}}<br />
==Problem 1==<br />
How many integer values of <math>x</math> satisfy <math>|x|<3\pi?</math><br />
<br />
<math>\textbf{(A) }9 \qquad \textbf{(B) }10 \qquad \textbf{(C) }18 \qquad \textbf{(D) }19 \qquad \textbf{(E) }20</math><br />
<br />
[[2021 AMC 12B Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
At a math contest, <math>57</math> students are wearing blue shirts, and another <math>75</math> students are wearing yellow shirts. The <math>132</math> students are assigned into <math>66</math> points. In exactly <math>23</math> of these pairs, both students are wearing blue shirts. In how many pairs are both studets wearing yellow shirts?<br />
<br />
<math>\textbf{(A) }23 \qquad \textbf{(B) }32 \qquad \textbf{(C) }37 \qquad \textbf{(D) }41 \qquad \textbf{(E) }64</math><br />
<br />
[[2021 AMC 12B Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
Suppose<cmath>2+\frac{1}{1+\frac{1}{2+\frac{2}{3+x}}}=\frac{144}{53}.</cmath>What is the value of <math>x?</math><br />
<br />
<math>\textbf{(A) }\frac34 \qquad \textbf{(B) }\frac78 \qquad \textbf{(C) }\frac{14}{15} \qquad \textbf{(D) }\frac{37}{38} \qquad \textbf{(E) }\frac{52}{53}</math><br />
<br />
[[2021 AMC 12B Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
Ms. Blackwell gives an exam to two classes. The mean of the scores of the students in the morning class is <math>84</math>, and the afternoon class's mean score is <math>70</math>. The ratio of the number of students in the morning clas to the number of students in the afternoon class is <math>\frac34</math>. What is the mean of the score of all the students?<br />
<br />
<math>\textbf{(A) }74 \qquad \textbf{(B) }75 \qquad \textbf{(C) }76 \qquad \textbf{(D) }77 \qquad \textbf{(E) }78</math><br />
<br />
[[2021 AMC 12B Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
The point <math>P(a,b)</math> in the <math>xy</math>-plane is first rotated counterclockwise by <math>90^\circ</math> around the point <math>(1,5)</math> and then reflected about the line <math>y=-x</math>. The image of <math>P</math> after these two transformations is at <math>(-6,3)</math>. What is <math>b-a?</math><br />
<br />
<math>\textbf{(A) }1 \qquad \textbf{(B) }3 \qquad \textbf{(C) }5 \qquad \textbf{(D) }7 \qquad \textbf{(E) }9</math><br />
<br />
[[2021 AMC 12B Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
An inverted cone with base radius <math>12 \text{cm}</math> and height <math>18\text{cm}</math> is full of water. The water is poured into a tall cylinder whose horizontal base has a radius of <math>24\text{cm}</math>. What is the height in centimeters of the water in the cylinder?<br />
<br />
<math>\textbf{(A) }1.5 \qquad \textbf{(B) }3 \qquad \textbf{(C) }4 \qquad \textbf{(D) }4.5 \qquad \textbf{(E) }6</math><br />
<br />
[[2021 AMC 12B Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
Let <math>N=34\cdot34\cdot63\cdot270.</math> What is the ratio of the sum of the odd divisors of <math>N</math> to the sum of the even divisors of <math>N?</math><br />
<br />
<math>\textbf{(A) }1:16 \qquad \textbf{(B) }1:15 \qquad \textbf{(C) }1:14 \qquad \textbf{(D) }1:8 \qquad \textbf{(E) }1:3</math><br />
<br />
[[2021 AMC 12B Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
Three equally spaced parallel lines intersect a circle, creating three chords of lengths <math>38,38,</math> and <math>34</math>. What is the distance between two adjacent parallel lines?<br />
<br />
<math>\textbf{(A) }5\frac12 \qquad \textbf{(B) }6 \qquad \textbf{(C) }6\frac12 \qquad \textbf{(D) }7 \qquad \textbf{(E) }7\frac12</math><br />
<br />
[[2021 AMC 12B Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
What is the value of<cmath>\frac{\log_2 80}{\log_{40}2}-\frac{\log_2 160}{\log_{20}2}?</cmath><br />
<math>\textbf{(A) }0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }\frac54 \qquad \textbf{(D) }2 \qquad \textbf{(E) }\log_2 5</math><br />
<br />
[[2021 AMC 12B Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
Two distinct numbers are selected from the set <math>\{1,2,3,4,\dots,36,37\}</math> so that the sum of the remaining <math>35</math> numbers is the product of these two numbers. What is the difference of these two numbers?<br />
<br />
<math>\textbf{(A) }5 \qquad \textbf{(B) }7 \qquad \textbf{(C) }8\qquad \textbf{(D) }9 \qquad \textbf{(E) }10</math><br />
<br />
[[2021 AMC 12B Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
Triangle <math>ABC</math> has <math>AB=13,BC=14</math> and <math>AC=15</math>. Let <math>P</math> be the point on <math>\overline{AC}</math> such that <math>PC=10</math>. There are exactly two points <math>D</math> and <math>E</math> on line <math>BP</math> such that quadrilaterals <math>ABCD</math> and <math>ABCE</math> are trapezoids. What is the distance <math>DE?</math><br />
<br />
<math>\textbf{(A) }\frac{42}5 \qquad \textbf{(B) }6\sqrt2 \qquad \textbf{(C) }\frac{84}5\qquad \textbf{(D) }12\sqrt2 \qquad \textbf{(E) }18</math><br />
<br />
[[2021 AMC 12B Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
Suppose that <math>S</math> is a finite set of positive integers. If the greatest integer in <math>S</math> is removed from <math>S</math>, then the average value (arithmetic mean) of the integers remaining is <math>32</math>. If the least integer in <math>S</math> is also removed, then the average value of the integers remaining is <math>35</math>. If the great integer is then returned to the set, the average value of the integers rises to <math>40.</math> The greatest integer in the original set <math>S</math> is <math>72</math> greater than the least integer in <math>S</math>. What is the average value of all the integers in the set <math>S?</math><br />
<br />
<math>\textbf{(A) }36.2 \qquad \textbf{(B) }36.4 \qquad \textbf{(C) }36.6\qquad \textbf{(D) }36.8 \qquad \textbf{(E) }37</math><br />
<br />
[[2021 AMC 12B Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
How many values of <math>\theta</math> in the interval <math>0<\theta\le 2\pi</math> satisfy<cmath>1-3\sin\theta+5\cos3\theta?</cmath><br />
<math>\textbf{(A) }2 \qquad \textbf{(B) }4 \qquad \textbf{(C) }5\qquad \textbf{(D) }6 \qquad \textbf{(E) }8</math><br />
<br />
[[2021 AMC 12B Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
Let <math>ABCD</math> be a rectangle and let <math>\overline{DM}</math> be a segment perpendicular to the plane of <math>ABCD</math>. Suppose that <math>\overline{DM}</math> has integer length, and the lengths of <math>\overline{MA},\overline{MC},</math> and <math>\overline{MB}</math> are consecutive odd positive integers (in this order). What is the volume of pyramid <math>MACD?</math><br />
<br />
<math>\textbf{(A) }24\sqrt5 \qquad \textbf{(B) }60 \qquad \textbf{(C) }28\sqrt5\qquad \textbf{(D) }66 \qquad \textbf{(E) }8\sqrt{70}</math><br />
<br />
[[2021 AMC 12B Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
The figure is constructed from <math>11</math> line segments, each of which has length <math>2</math>. The area of pentagon <math>ABCDE</math> can be written is <math>\sqrt{m} + \sqrt{n}</math>, where <math>m</math> and <math>n</math> are positive integers. What is <math>m + n ?</math><br />
<asy> /* Made by samrocksnature */ pair A=(-2.4638,4.10658); pair B=(-4,2.6567453480756127); pair C=(-3.47132,0.6335248637894945); pair D=(-1.464483379039766,0.6335248637894945); pair E=(-0.956630463955801,2.6567453480756127); pair F=(-2,2); pair G=(-3,2); draw(A--B--C--D--E--A); draw(A--F--A--G); draw(B--F--C); draw(E--G--D); label("A",A,N); label("B",B,W); label("C",C,W); label("D",D,E); label("E",E,dir(0)); dot(A^^B^^C^^D^^E^^F^^G); </asy><br />
<math>\textbf{(A)} ~20 \qquad\textbf{(B)} ~21 \qquad\textbf{(C)} ~22 \qquad\textbf{(D)} ~23 \qquad\textbf{(E)} ~24</math><br />
<br />
[[2021 AMC 12B Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
Let <math>g(x)</math> be a polynomial with leading coefficient <math>1,</math> whose three roots are the reciprocals of the three roots of <math>f(x)=x^3+ax^2+bx+c,</math> where <math>1<a<b<c.</math> What is <math>g(1)</math> in terms of <math>a,b,</math> and <math>c?</math><br />
<br />
<math>\textbf{(A) }\frac{1+a+b+c}c \qquad \textbf{(B) }1+a+b+c \qquad \textbf{(C) }\frac{1+a+b+c}{c^2}\qquad \textbf{(D) }\frac{a+b+c}{c^2} \qquad \textbf{(E) }\frac{1+a+b+c}{a+b+c}</math><br />
<br />
[[2021 AMC 12B Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
Let <math>ABCD</math> be an isoceles trapezoid having parallel bases <math>\overline{AB}</math> and <math>\overline{CD}</math> with <math>AB>CD.</math> Line segments from a point inside <math>ABCD</math> to the vertices divide the trapezoid into four triangles whose areas are <math>2, 3, 4,</math> and <math>5</math> starting with the triangle with base <math>\overline{CD}</math> and moving clockwise as shown in the diagram below. What is the ratio <math>\frac{AB}{CD}?</math><br />
<asy><br />
unitsize(100);<br />
pair A=(-1, 0), B=(1, 0), C=(0.3, 0.9), D=(-0.3, 0.9), P=(0.2, 0.5), E=(0.1, 0.75), F=(0.4, 0.5), G=(0.15, 0.2), H=(-0.3, 0.5); <br />
draw(A--B--C--D--cycle, black); <br />
draw(A--P, black);<br />
draw(B--P, black);<br />
draw(C--P, black);<br />
draw(D--P, black);<br />
label("$A$",A,(-1,0));<br />
label("$B$",B,(1,0));<br />
label("$C$",C,(1,-0));<br />
label("$D$",D,(-1,0));<br />
label("$2$",E,(0,0));<br />
label("$3$",F,(0,0));<br />
label("$4$",G,(0,0));<br />
label("$5$",H,(0,0));<br />
dot(A^^B^^C^^D^^P);<br />
</asy><br />
<math>\textbf{(A)}\: 3\qquad\textbf{(B)}\: 2+\sqrt{2}\qquad\textbf{(C)}\: 1+\sqrt{6}\qquad\textbf{(D)}\: 2\sqrt{3}\qquad\textbf{(E)}\: 3\sqrt{2}</math><br />
<br />
[[2021 AMC 12B Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
Let <math>z</math> be a complex number satisfying <math>12|z|^2=2|z+2|^2+|z^2+1|^2+31.</math> What is the value of <math>z+\frac 6z?</math><br />
<br />
<math>\textbf{(A) }-2 \qquad \textbf{(B) }-1 \qquad \textbf{(C) }\frac12\qquad \textbf{(D) }1 \qquad \textbf{(E) }4</math><br />
<br />
[[2021 AMC 12B Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
Two fair dice, each with at least <math>6</math> faces are rolled. On each face of each dice is printed a distinct integer from <math>1</math> to the number of faces on that die, inclusive. The probability of rolling a sum if <math>7</math> is <math>\frac34</math> of the probability of rolling a sum of <math>10,</math> and the probability of rolling a sum of <math>12</math> is <math>\frac{1}{12}</math>. What is the least possible number of faces on the two dice combined?<br />
<br />
<math>\textbf{(A) }16 \qquad \textbf{(B) }17 \qquad \textbf{(C) }18\qquad \textbf{(D) }19 \qquad \textbf{(E) }20</math><br />
<br />
[[2021 AMC 12B Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
Let <math>Q(z)</math> and <math>R(z)</math> be the unique polynomials such that<cmath>z^{2021}+1=(z^2+z+1)Q(z)+R(z)</cmath>and the degree of <math>R</math> is less than <math>2.</math> What is <math>R(z)?</math><br />
<br />
<math>\textbf{(A) }-z \qquad \textbf{(B) }-1 \qquad \textbf{(C) }2021\qquad \textbf{(D) }z+1 \qquad \textbf{(E) }2z+1</math><br />
<br />
[[2021 AMC 12B Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
Let <math>S</math> be the sum of all positive real numbers <math>x</math> for which<cmath>x^{2^{\sqrt2}}=\sqrt2^{2^x}.</cmath>Which of the following statements is true?<br />
<br />
<math>\textbf{(A) }S<\sqrt2 \qquad \textbf{(B) }S=\sqrt2 \qquad \textbf{(C) }\sqrt2<S<2\qquad \textbf{(D) }2\le S<6 \qquad \textbf{(E) }S\ge 6</math><br />
<br />
[[2021 AMC 12B Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
Arjun and Beth play a game in which they take turns removing one brick or two adjacent bricks from one "wall" among a set of several walls of bricks, with gaps possibly creating new walls. The walls are one brick tall. For example, a set of walls of sizes <math>4</math> and <math>2</math> can be changed into any of the following by one move: <math>(3,2),(2,1,2),(4),(4,1),(2,2),</math> or <math>(1,1,2).</math> <br />
<br />
<asy><br />
unitsize(4mm);<br />
real[] boxes = {0,1,2,3,5,6,13,14,15,17,18,21,22,24,26,27,30,31,32,33};<br />
for(real i:boxes){<br />
draw(box((i,0),(i+1,3)));<br />
}<br />
draw((8,1.5)--(12,1.5),Arrow());<br />
defaultpen(fontsize(20pt));<br />
label(",",(20,0));<br />
label(",",(29,0));<br />
label(",...",(35.5,0));<br />
</asy><br />
<br />
Arjun plays first, and the player who removes the last brick wins. For which starting configuration is there a strategy that guarantees a win for Beth?<br />
<br />
<math>\textbf{(A) }(6,1,1) \qquad \textbf{(B) }(6,2,1) \qquad \textbf{(C) }(6,2,2)\qquad \textbf{(D) }(6,3,1) \qquad \textbf{(E) }(6,3,2)</math><br />
<br />
[[2021 AMC 12B Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
Three balls are randomly and independantly tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin <math>i</math> is <math>2^{-i}</math> for <math>i=1,2,3,....</math> More than one ball is allowed in each bin. The probability that the balls end up evenly spaced in distinct bins is <math>\frac pq,</math> where <math>p</math> and <math>q</math> are relatively prime positive integers. (For example, the balls are evenly spaced if they are tossed into bins <math>3,17,</math> and <math>10.</math>) What is <math>p+q?</math><br />
<br />
<math>\textbf{(A) }55 \qquad \textbf{(B) }56 \qquad \textbf{(C) }57\qquad \textbf{(D) }58 \qquad \textbf{(E) }59</math><br />
<br />
[[2021 AMC 12B Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
Let <math>ABCD</math> be a parallelogram with area <math>15</math>. Points <math>P</math> and <math>Q</math> are the projections of <math>A</math> and <math>C,</math> respectively, onto the line <math>BD;</math> and points <math>R</math> and <math>S</math> are the projections of <math>B</math> and <math>D,</math> respectively, onto the line <math>AC.</math> See the figure, which also shows the relative locations of these points.<br />
<br />
<asy><br />
size(350);<br />
defaultpen(linewidth(0.8)+fontsize(11));<br />
real theta = aTan(1.25/2);<br />
pair A = 2.5*dir(180+theta), B = (3.35,0), C = -A, D = -B, P = foot(A,B,D), Q = -P, R = foot(B,A,C), S = -R;<br />
draw(A--B--C--D--A^^B--D^^R--S^^rightanglemark(A,P,D,6)^^rightanglemark(C,Q,D,6));<br />
draw(B--R^^C--Q^^A--P^^D--S,linetype("4 4"));<br />
dot("$A$",A,dir(270));<br />
dot("$B$",B,E);<br />
dot("$C$",C,N);<br />
dot("$D$",D,W);<br />
dot("$P$",P,SE);<br />
dot("$Q$",Q,NE);<br />
dot("$R$",R,N);<br />
dot("$S$",S,dir(270));<br />
</asy><br />
<br />
Suppose <math>PQ=6</math> and <math>RS=8,</math> and let <math>d</math> denote the length of <math>\overline{BD},</math> the longer diagonal of <math>ABCD.</math> Then <math>d^2</math> can be written in the form <math>m+n\sqrt p,</math> where <math>m,n,</math> and <math>p</math> are positive integers and <math>p</math> is not divisible by the square of any prime. What is <math>m+n+p?</math><br />
<br />
<math>\textbf{(A) }81 \qquad \textbf{(B) }89 \qquad \textbf{(C) }97\qquad \textbf{(D) }105 \qquad \textbf{(E) }113</math><br />
<br />
[[2021 AMC 12B Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
Let <math>S</math> be the set of lattice points in the coordinate plane, both of whose coordinates are integers between <math>1</math> and <math>30,</math> inclusive. Exactly <math>300</math> points in <math>S</math> lie on or below a line with equation <math>y=mx.</math> The possible values of <math>m</math> lie in an interval of length <math>\frac ab,</math> where <math>a</math> and <math>b</math> are relatively prime positive integers. What is <math>a+b?</math><br />
<br />
<math>\textbf{(A) }31 \qquad \textbf{(B) }47 \qquad \textbf{(C) }62\qquad \textbf{(D) }72 \qquad \textbf{(E) }85</math><br />
<br />
[[2021 AMC 12B Problems/Problem 25|Solution]]<br />
<br />
<br />
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 900 lattice points. I estimate that the slope, m, 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 600 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 are is above the line. <br />
<br />
I find that the number of coordinates with <math>x=1</math> above the line is 30, 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 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 <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 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. <br />
<br />
To find the upper bound, notice that each point with an integer-valued x-coordinate is either <math>\frac{1}{3}</math> or <math>\frac{2}{3}</math> 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 <math>x=28, 29, 30</math> which the line <math>y=\frac{2}{3}x</math> intersects at <math>y= \frac{56}{3}, \frac{58}{3}, 20</math>. 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, <math>\frac{56}{3}</math>) is closer to the lattice point. The slope of the line which goes through both the origin and (28,19) is <math>y=\frac{19}{28}x</math>. This gives an upper bound of <math>m=\frac{19}{28}</math>. <br />
<br />
Taking the upper bound of m and subtracting the lower bound yields <math>\frac{19}{28}-\frac{2}{3}=\frac{1}{84}</math>. This is answer E.<br />
<br />
<br />
~Theajl<br />
<br />
==See also==<br />
{{AMC12 box|year=2021|ab=B|before=[[2021 AMC 12A Problems]]|after=[[2022 AMC 12A Problems]]}}<br />
<br />
[[Category:AMC 12 Problems]]<br />
{{MAA Notice}}</div>Theajlhttps://artofproblemsolving.com/wiki/index.php?title=2021_AMC_12B_Problems&diff=1457162021 AMC 12B Problems2021-02-12T03:56:25Z<p>Theajl: /* Problem 25 */</p>
<hr />
<div>{{AMC12 Problems|year=2021|ab=B}}<br />
==Problem 1==<br />
How many integer values of <math>x</math> satisfy <math>|x|<3\pi?</math><br />
<br />
<math>\textbf{(A) }9 \qquad \textbf{(B) }10 \qquad \textbf{(C) }18 \qquad \textbf{(D) }19 \qquad \textbf{(E) }20</math><br />
<br />
[[2021 AMC 12B Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
At a math contest, <math>57</math> students are wearing blue shirts, and another <math>75</math> students are wearing yellow shirts. The <math>132</math> students are assigned into <math>66</math> points. In exactly <math>23</math> of these pairs, both students are wearing blue shirts. In how many pairs are both studets wearing yellow shirts?<br />
<br />
<math>\textbf{(A) }23 \qquad \textbf{(B) }32 \qquad \textbf{(C) }37 \qquad \textbf{(D) }41 \qquad \textbf{(E) }64</math><br />
<br />
[[2021 AMC 12B Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
Suppose<cmath>2+\frac{1}{1+\frac{1}{2+\frac{2}{3+x}}}=\frac{144}{53}.</cmath>What is the value of <math>x?</math><br />
<br />
<math>\textbf{(A) }\frac34 \qquad \textbf{(B) }\frac78 \qquad \textbf{(C) }\frac{14}{15} \qquad \textbf{(D) }\frac{37}{38} \qquad \textbf{(E) }\frac{52}{53}</math><br />
<br />
[[2021 AMC 12B Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
Ms. Blackwell gives an exam to two classes. The mean of the scores of the students in the morning class is <math>84</math>, and the afternoon class's mean score is <math>70</math>. The ratio of the number of students in the morning clas to the number of students in the afternoon class is <math>\frac34</math>. What is the mean of the score of all the students?<br />
<br />
<math>\textbf{(A) }74 \qquad \textbf{(B) }75 \qquad \textbf{(C) }76 \qquad \textbf{(D) }77 \qquad \textbf{(E) }78</math><br />
<br />
[[2021 AMC 12B Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
The point <math>P(a,b)</math> in the <math>xy</math>-plane is first rotated counterclockwise by <math>90^\circ</math> around the point <math>(1,5)</math> and then reflected about the line <math>y=-x</math>. The image of <math>P</math> after these two transformations is at <math>(-6,3)</math>. What is <math>b-a?</math><br />
<br />
<math>\textbf{(A) }1 \qquad \textbf{(B) }3 \qquad \textbf{(C) }5 \qquad \textbf{(D) }7 \qquad \textbf{(E) }9</math><br />
<br />
[[2021 AMC 12B Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
An inverted cone with base radius <math>12 \text{cm}</math> and height <math>18\text{cm}</math> is full of water. The water is poured into a tall cylinder whose horizontal base has a radius of <math>24\text{cm}</math>. What is the height in centimeters of the water in the cylinder?<br />
<br />
<math>\textbf{(A) }1.5 \qquad \textbf{(B) }3 \qquad \textbf{(C) }4 \qquad \textbf{(D) }4.5 \qquad \textbf{(E) }6</math><br />
<br />
[[2021 AMC 12B Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
Let <math>N=34\cdot34\cdot63\cdot270.</math> What is the ratio of the sum of the odd divisors of <math>N</math> to the sum of the even divisors of <math>N?</math><br />
<br />
<math>\textbf{(A) }1:16 \qquad \textbf{(B) }1:15 \qquad \textbf{(C) }1:14 \qquad \textbf{(D) }1:8 \qquad \textbf{(E) }1:3</math><br />
<br />
[[2021 AMC 12B Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
Three equally spaced parallel lines intersect a circle, creating three chords of lengths <math>38,38,</math> and <math>34</math>. What is the distance between two adjacent parallel lines?<br />
<br />
<math>\textbf{(A) }5\frac12 \qquad \textbf{(B) }6 \qquad \textbf{(C) }6\frac12 \qquad \textbf{(D) }7 \qquad \textbf{(E) }7\frac12</math><br />
<br />
[[2021 AMC 12B Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
What is the value of<cmath>\frac{\log_2 80}{\log_{40}2}-\frac{\log_2 160}{\log_{20}2}?</cmath><br />
<math>\textbf{(A) }0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }\frac54 \qquad \textbf{(D) }2 \qquad \textbf{(E) }\log_2 5</math><br />
<br />
[[2021 AMC 12B Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
Two distinct numbers are selected from the set <math>\{1,2,3,4,\dots,36,37\}</math> so that the sum of the remaining <math>35</math> numbers is the product of these two numbers. What is the difference of these two numbers?<br />
<br />
<math>\textbf{(A) }5 \qquad \textbf{(B) }7 \qquad \textbf{(C) }8\qquad \textbf{(D) }9 \qquad \textbf{(E) }10</math><br />
<br />
[[2021 AMC 12B Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
Triangle <math>ABC</math> has <math>AB=13,BC=14</math> and <math>AC=15</math>. Let <math>P</math> be the point on <math>\overline{AC}</math> such that <math>PC=10</math>. There are exactly two points <math>D</math> and <math>E</math> on line <math>BP</math> such that quadrilaterals <math>ABCD</math> and <math>ABCE</math> are trapezoids. What is the distance <math>DE?</math><br />
<br />
<math>\textbf{(A) }\frac{42}5 \qquad \textbf{(B) }6\sqrt2 \qquad \textbf{(C) }\frac{84}5\qquad \textbf{(D) }12\sqrt2 \qquad \textbf{(E) }18</math><br />
<br />
[[2021 AMC 12B Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
Suppose that <math>S</math> is a finite set of positive integers. If the greatest integer in <math>S</math> is removed from <math>S</math>, then the average value (arithmetic mean) of the integers remaining is <math>32</math>. If the least integer in <math>S</math> is also removed, then the average value of the integers remaining is <math>35</math>. If the great integer is then returned to the set, the average value of the integers rises to <math>40.</math> The greatest integer in the original set <math>S</math> is <math>72</math> greater than the least integer in <math>S</math>. What is the average value of all the integers in the set <math>S?</math><br />
<br />
<math>\textbf{(A) }36.2 \qquad \textbf{(B) }36.4 \qquad \textbf{(C) }36.6\qquad \textbf{(D) }36.8 \qquad \textbf{(E) }37</math><br />
<br />
[[2021 AMC 12B Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
How many values of <math>\theta</math> in the interval <math>0<\theta\le 2\pi</math> satisfy<cmath>1-3\sin\theta+5\cos3\theta?</cmath><br />
<math>\textbf{(A) }2 \qquad \textbf{(B) }4 \qquad \textbf{(C) }5\qquad \textbf{(D) }6 \qquad \textbf{(E) }8</math><br />
<br />
[[2021 AMC 12B Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
Let <math>ABCD</math> be a rectangle and let <math>\overline{DM}</math> be a segment perpendicular to the plane of <math>ABCD</math>. Suppose that <math>\overline{DM}</math> has integer length, and the lengths of <math>\overline{MA},\overline{MC},</math> and <math>\overline{MB}</math> are consecutive odd positive integers (in this order). What is the volume of pyramid <math>MACD?</math><br />
<br />
<math>\textbf{(A) }24\sqrt5 \qquad \textbf{(B) }60 \qquad \textbf{(C) }28\sqrt5\qquad \textbf{(D) }66 \qquad \textbf{(E) }8\sqrt{70}</math><br />
<br />
[[2021 AMC 12B Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
The figure is constructed from <math>11</math> line segments, each of which has length <math>2</math>. The area of pentagon <math>ABCDE</math> can be written is <math>\sqrt{m} + \sqrt{n}</math>, where <math>m</math> and <math>n</math> are positive integers. What is <math>m + n ?</math><br />
<asy> /* Made by samrocksnature */ pair A=(-2.4638,4.10658); pair B=(-4,2.6567453480756127); pair C=(-3.47132,0.6335248637894945); pair D=(-1.464483379039766,0.6335248637894945); pair E=(-0.956630463955801,2.6567453480756127); pair F=(-2,2); pair G=(-3,2); draw(A--B--C--D--E--A); draw(A--F--A--G); draw(B--F--C); draw(E--G--D); label("A",A,N); label("B",B,W); label("C",C,W); label("D",D,E); label("E",E,dir(0)); dot(A^^B^^C^^D^^E^^F^^G); </asy><br />
<math>\textbf{(A)} ~20 \qquad\textbf{(B)} ~21 \qquad\textbf{(C)} ~22 \qquad\textbf{(D)} ~23 \qquad\textbf{(E)} ~24</math><br />
<br />
[[2021 AMC 12B Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
Let <math>g(x)</math> be a polynomial with leading coefficient <math>1,</math> whose three roots are the reciprocals of the three roots of <math>f(x)=x^3+ax^2+bx+c,</math> where <math>1<a<b<c.</math> What is <math>g(1)</math> in terms of <math>a,b,</math> and <math>c?</math><br />
<br />
<math>\textbf{(A) }\frac{1+a+b+c}c \qquad \textbf{(B) }1+a+b+c \qquad \textbf{(C) }\frac{1+a+b+c}{c^2}\qquad \textbf{(D) }\frac{a+b+c}{c^2} \qquad \textbf{(E) }\frac{1+a+b+c}{a+b+c}</math><br />
<br />
[[2021 AMC 12B Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
Let <math>ABCD</math> be an isoceles trapezoid having parallel bases <math>\overline{AB}</math> and <math>\overline{CD}</math> with <math>AB>CD.</math> Line segments from a point inside <math>ABCD</math> to the vertices divide the trapezoid into four triangles whose areas are <math>2, 3, 4,</math> and <math>5</math> starting with the triangle with base <math>\overline{CD}</math> and moving clockwise as shown in the diagram below. What is the ratio <math>\frac{AB}{CD}?</math><br />
<asy><br />
unitsize(100);<br />
pair A=(-1, 0), B=(1, 0), C=(0.3, 0.9), D=(-0.3, 0.9), P=(0.2, 0.5), E=(0.1, 0.75), F=(0.4, 0.5), G=(0.15, 0.2), H=(-0.3, 0.5); <br />
draw(A--B--C--D--cycle, black); <br />
draw(A--P, black);<br />
draw(B--P, black);<br />
draw(C--P, black);<br />
draw(D--P, black);<br />
label("$A$",A,(-1,0));<br />
label("$B$",B,(1,0));<br />
label("$C$",C,(1,-0));<br />
label("$D$",D,(-1,0));<br />
label("$2$",E,(0,0));<br />
label("$3$",F,(0,0));<br />
label("$4$",G,(0,0));<br />
label("$5$",H,(0,0));<br />
dot(A^^B^^C^^D^^P);<br />
</asy><br />
<math>\textbf{(A)}\: 3\qquad\textbf{(B)}\: 2+\sqrt{2}\qquad\textbf{(C)}\: 1+\sqrt{6}\qquad\textbf{(D)}\: 2\sqrt{3}\qquad\textbf{(E)}\: 3\sqrt{2}</math><br />
<br />
[[2021 AMC 12B Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
Let <math>z</math> be a complex number satisfying <math>12|z|^2=2|z+2|^2+|z^2+1|^2+31.</math> What is the value of <math>z+\frac 6z?</math><br />
<br />
<math>\textbf{(A) }-2 \qquad \textbf{(B) }-1 \qquad \textbf{(C) }\frac12\qquad \textbf{(D) }1 \qquad \textbf{(E) }4</math><br />
<br />
[[2021 AMC 12B Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
Two fair dice, each with at least <math>6</math> faces are rolled. On each face of each dice is printed a distinct integer from <math>1</math> to the number of faces on that die, inclusive. The probability of rolling a sum if <math>7</math> is <math>\frac34</math> of the probability of rolling a sum of <math>10,</math> and the probability of rolling a sum of <math>12</math> is <math>\frac{1}{12}</math>. What is the least possible number of faces on the two dice combined?<br />
<br />
<math>\textbf{(A) }16 \qquad \textbf{(B) }17 \qquad \textbf{(C) }18\qquad \textbf{(D) }19 \qquad \textbf{(E) }20</math><br />
<br />
[[2021 AMC 12B Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
Let <math>Q(z)</math> and <math>R(z)</math> be the unique polynomials such that<cmath>z^{2021}+1=(z^2+z+1)Q(z)+R(z)</cmath>and the degree of <math>R</math> is less than <math>2.</math> What is <math>R(z)?</math><br />
<br />
<math>\textbf{(A) }-z \qquad \textbf{(B) }-1 \qquad \textbf{(C) }2021\qquad \textbf{(D) }z+1 \qquad \textbf{(E) }2z+1</math><br />
<br />
[[2021 AMC 12B Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
Let <math>S</math> be the sum of all positive real numbers <math>x</math> for which<cmath>x^{2^{\sqrt2}}=\sqrt2^{2^x}.</cmath>Which of the following statements is true?<br />
<br />
<math>\textbf{(A) }S<\sqrt2 \qquad \textbf{(B) }S=\sqrt2 \qquad \textbf{(C) }\sqrt2<S<2\qquad \textbf{(D) }2\le S<6 \qquad \textbf{(E) }S\ge 6</math><br />
<br />
[[2021 AMC 12B Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
Arjun and Beth play a game in which they take turns removing one brick or two adjacent bricks from one "wall" among a set of several walls of bricks, with gaps possibly creating new walls. The walls are one brick tall. For example, a set of walls of sizes <math>4</math> and <math>2</math> can be changed into any of the following by one move: <math>(3,2),(2,1,2),(4),(4,1),(2,2),</math> or <math>(1,1,2).</math> <br />
<br />
<asy><br />
unitsize(4mm);<br />
real[] boxes = {0,1,2,3,5,6,13,14,15,17,18,21,22,24,26,27,30,31,32,33};<br />
for(real i:boxes){<br />
draw(box((i,0),(i+1,3)));<br />
}<br />
draw((8,1.5)--(12,1.5),Arrow());<br />
defaultpen(fontsize(20pt));<br />
label(",",(20,0));<br />
label(",",(29,0));<br />
label(",...",(35.5,0));<br />
</asy><br />
<br />
Arjun plays first, and the player who removes the last brick wins. For which starting configuration is there a strategy that guarantees a win for Beth?<br />
<br />
<math>\textbf{(A) }(6,1,1) \qquad \textbf{(B) }(6,2,1) \qquad \textbf{(C) }(6,2,2)\qquad \textbf{(D) }(6,3,1) \qquad \textbf{(E) }(6,3,2)</math><br />
<br />
[[2021 AMC 12B Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
Three balls are randomly and independantly tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin <math>i</math> is <math>2^{-i}</math> for <math>i=1,2,3,....</math> More than one ball is allowed in each bin. The probability that the balls end up evenly spaced in distinct bins is <math>\frac pq,</math> where <math>p</math> and <math>q</math> are relatively prime positive integers. (For example, the balls are evenly spaced if they are tossed into bins <math>3,17,</math> and <math>10.</math>) What is <math>p+q?</math><br />
<br />
<math>\textbf{(A) }55 \qquad \textbf{(B) }56 \qquad \textbf{(C) }57\qquad \textbf{(D) }58 \qquad \textbf{(E) }59</math><br />
<br />
[[2021 AMC 12B Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
Let <math>ABCD</math> be a parallelogram with area <math>15</math>. Points <math>P</math> and <math>Q</math> are the projections of <math>A</math> and <math>C,</math> respectively, onto the line <math>BD;</math> and points <math>R</math> and <math>S</math> are the projections of <math>B</math> and <math>D,</math> respectively, onto the line <math>AC.</math> See the figure, which also shows the relative locations of these points.<br />
<br />
<asy><br />
size(350);<br />
defaultpen(linewidth(0.8)+fontsize(11));<br />
real theta = aTan(1.25/2);<br />
pair A = 2.5*dir(180+theta), B = (3.35,0), C = -A, D = -B, P = foot(A,B,D), Q = -P, R = foot(B,A,C), S = -R;<br />
draw(A--B--C--D--A^^B--D^^R--S^^rightanglemark(A,P,D,6)^^rightanglemark(C,Q,D,6));<br />
draw(B--R^^C--Q^^A--P^^D--S,linetype("4 4"));<br />
dot("$A$",A,dir(270));<br />
dot("$B$",B,E);<br />
dot("$C$",C,N);<br />
dot("$D$",D,W);<br />
dot("$P$",P,SE);<br />
dot("$Q$",Q,NE);<br />
dot("$R$",R,N);<br />
dot("$S$",S,dir(270));<br />
</asy><br />
<br />
Suppose <math>PQ=6</math> and <math>RS=8,</math> and let <math>d</math> denote the length of <math>\overline{BD},</math> the longer diagonal of <math>ABCD.</math> Then <math>d^2</math> can be written in the form <math>m+n\sqrt p,</math> where <math>m,n,</math> and <math>p</math> are positive integers and <math>p</math> is not divisible by the square of any prime. What is <math>m+n+p?</math><br />
<br />
<math>\textbf{(A) }81 \qquad \textbf{(B) }89 \qquad \textbf{(C) }97\qquad \textbf{(D) }105 \qquad \textbf{(E) }113</math><br />
<br />
[[2021 AMC 12B Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
Let <math>S</math> be the set of lattice points in the coordinate plane, both of whose coordinates are integers between <math>1</math> and <math>30,</math> inclusive. Exactly <math>300</math> points in <math>S</math> lie on or below a line with equation <math>y=mx.</math> The possible values of <math>m</math> lie in an interval of length <math>\frac ab,</math> where <math>a</math> and <math>b</math> are relatively prime positive integers. What is <math>a+b?</math><br />
<br />
<math>\textbf{(A) }31 \qquad \textbf{(B) }47 \qquad \textbf{(C) }62\qquad \textbf{(D) }72 \qquad \textbf{(E) }85</math><br />
<br />
[[2021 AMC 12B Problems/Problem 25|Solution]]<br />
<br />
<br />
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 900 lattice points. I estimate that the slope, m, 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 600 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 are is above the line. <br />
<br />
I find that the number of coordinates with <math>x=1</math> above the line is 30, 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 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 <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 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. <br />
<br />
To find the upper bound, notice that each point with an integer-valued x-coordinate is either <math>\frac{1}{3}</math> or <math>\frac{2}{3}</math> 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 <math>x=28, 29, 30</math> which the line <math>y=\frac{2}{3}x</math> intersects at <math>y= \frac{56}{3}, \frac{58}{3}, 20</math>. 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, <math>\frac{56}{3}</math>) is closer to the lattice point. The slope of the line which goes through both the origin and (28,19) is <math>y=\frac{19}{28}x</math>. This gives an upper bound of <math>m=\frac{19}{28}</math>. <br />
<br />
Taking the upper bound of m and subtracting the lower bound yields <math>\frac{19}{28}-\frac{2}{3}=\frac{1}{84}</math>. This is answer E.<br />
<br />
==See also==<br />
{{AMC12 box|year=2021|ab=B|before=[[2021 AMC 12A Problems]]|after=[[2022 AMC 12A Problems]]}}<br />
<br />
[[Category:AMC 12 Problems]]<br />
{{MAA Notice}}</div>Theajlhttps://artofproblemsolving.com/wiki/index.php?title=2021_AMC_12B_Problems&diff=1457152021 AMC 12B Problems2021-02-12T03:55:43Z<p>Theajl: /* Problem 25 */</p>
<hr />
<div>{{AMC12 Problems|year=2021|ab=B}}<br />
==Problem 1==<br />
How many integer values of <math>x</math> satisfy <math>|x|<3\pi?</math><br />
<br />
<math>\textbf{(A) }9 \qquad \textbf{(B) }10 \qquad \textbf{(C) }18 \qquad \textbf{(D) }19 \qquad \textbf{(E) }20</math><br />
<br />
[[2021 AMC 12B Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
At a math contest, <math>57</math> students are wearing blue shirts, and another <math>75</math> students are wearing yellow shirts. The <math>132</math> students are assigned into <math>66</math> points. In exactly <math>23</math> of these pairs, both students are wearing blue shirts. In how many pairs are both studets wearing yellow shirts?<br />
<br />
<math>\textbf{(A) }23 \qquad \textbf{(B) }32 \qquad \textbf{(C) }37 \qquad \textbf{(D) }41 \qquad \textbf{(E) }64</math><br />
<br />
[[2021 AMC 12B Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
Suppose<cmath>2+\frac{1}{1+\frac{1}{2+\frac{2}{3+x}}}=\frac{144}{53}.</cmath>What is the value of <math>x?</math><br />
<br />
<math>\textbf{(A) }\frac34 \qquad \textbf{(B) }\frac78 \qquad \textbf{(C) }\frac{14}{15} \qquad \textbf{(D) }\frac{37}{38} \qquad \textbf{(E) }\frac{52}{53}</math><br />
<br />
[[2021 AMC 12B Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
Ms. Blackwell gives an exam to two classes. The mean of the scores of the students in the morning class is <math>84</math>, and the afternoon class's mean score is <math>70</math>. The ratio of the number of students in the morning clas to the number of students in the afternoon class is <math>\frac34</math>. What is the mean of the score of all the students?<br />
<br />
<math>\textbf{(A) }74 \qquad \textbf{(B) }75 \qquad \textbf{(C) }76 \qquad \textbf{(D) }77 \qquad \textbf{(E) }78</math><br />
<br />
[[2021 AMC 12B Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
The point <math>P(a,b)</math> in the <math>xy</math>-plane is first rotated counterclockwise by <math>90^\circ</math> around the point <math>(1,5)</math> and then reflected about the line <math>y=-x</math>. The image of <math>P</math> after these two transformations is at <math>(-6,3)</math>. What is <math>b-a?</math><br />
<br />
<math>\textbf{(A) }1 \qquad \textbf{(B) }3 \qquad \textbf{(C) }5 \qquad \textbf{(D) }7 \qquad \textbf{(E) }9</math><br />
<br />
[[2021 AMC 12B Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
An inverted cone with base radius <math>12 \text{cm}</math> and height <math>18\text{cm}</math> is full of water. The water is poured into a tall cylinder whose horizontal base has a radius of <math>24\text{cm}</math>. What is the height in centimeters of the water in the cylinder?<br />
<br />
<math>\textbf{(A) }1.5 \qquad \textbf{(B) }3 \qquad \textbf{(C) }4 \qquad \textbf{(D) }4.5 \qquad \textbf{(E) }6</math><br />
<br />
[[2021 AMC 12B Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
Let <math>N=34\cdot34\cdot63\cdot270.</math> What is the ratio of the sum of the odd divisors of <math>N</math> to the sum of the even divisors of <math>N?</math><br />
<br />
<math>\textbf{(A) }1:16 \qquad \textbf{(B) }1:15 \qquad \textbf{(C) }1:14 \qquad \textbf{(D) }1:8 \qquad \textbf{(E) }1:3</math><br />
<br />
[[2021 AMC 12B Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
Three equally spaced parallel lines intersect a circle, creating three chords of lengths <math>38,38,</math> and <math>34</math>. What is the distance between two adjacent parallel lines?<br />
<br />
<math>\textbf{(A) }5\frac12 \qquad \textbf{(B) }6 \qquad \textbf{(C) }6\frac12 \qquad \textbf{(D) }7 \qquad \textbf{(E) }7\frac12</math><br />
<br />
[[2021 AMC 12B Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
What is the value of<cmath>\frac{\log_2 80}{\log_{40}2}-\frac{\log_2 160}{\log_{20}2}?</cmath><br />
<math>\textbf{(A) }0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }\frac54 \qquad \textbf{(D) }2 \qquad \textbf{(E) }\log_2 5</math><br />
<br />
[[2021 AMC 12B Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
Two distinct numbers are selected from the set <math>\{1,2,3,4,\dots,36,37\}</math> so that the sum of the remaining <math>35</math> numbers is the product of these two numbers. What is the difference of these two numbers?<br />
<br />
<math>\textbf{(A) }5 \qquad \textbf{(B) }7 \qquad \textbf{(C) }8\qquad \textbf{(D) }9 \qquad \textbf{(E) }10</math><br />
<br />
[[2021 AMC 12B Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
Triangle <math>ABC</math> has <math>AB=13,BC=14</math> and <math>AC=15</math>. Let <math>P</math> be the point on <math>\overline{AC}</math> such that <math>PC=10</math>. There are exactly two points <math>D</math> and <math>E</math> on line <math>BP</math> such that quadrilaterals <math>ABCD</math> and <math>ABCE</math> are trapezoids. What is the distance <math>DE?</math><br />
<br />
<math>\textbf{(A) }\frac{42}5 \qquad \textbf{(B) }6\sqrt2 \qquad \textbf{(C) }\frac{84}5\qquad \textbf{(D) }12\sqrt2 \qquad \textbf{(E) }18</math><br />
<br />
[[2021 AMC 12B Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
Suppose that <math>S</math> is a finite set of positive integers. If the greatest integer in <math>S</math> is removed from <math>S</math>, then the average value (arithmetic mean) of the integers remaining is <math>32</math>. If the least integer in <math>S</math> is also removed, then the average value of the integers remaining is <math>35</math>. If the great integer is then returned to the set, the average value of the integers rises to <math>40.</math> The greatest integer in the original set <math>S</math> is <math>72</math> greater than the least integer in <math>S</math>. What is the average value of all the integers in the set <math>S?</math><br />
<br />
<math>\textbf{(A) }36.2 \qquad \textbf{(B) }36.4 \qquad \textbf{(C) }36.6\qquad \textbf{(D) }36.8 \qquad \textbf{(E) }37</math><br />
<br />
[[2021 AMC 12B Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
How many values of <math>\theta</math> in the interval <math>0<\theta\le 2\pi</math> satisfy<cmath>1-3\sin\theta+5\cos3\theta?</cmath><br />
<math>\textbf{(A) }2 \qquad \textbf{(B) }4 \qquad \textbf{(C) }5\qquad \textbf{(D) }6 \qquad \textbf{(E) }8</math><br />
<br />
[[2021 AMC 12B Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
Let <math>ABCD</math> be a rectangle and let <math>\overline{DM}</math> be a segment perpendicular to the plane of <math>ABCD</math>. Suppose that <math>\overline{DM}</math> has integer length, and the lengths of <math>\overline{MA},\overline{MC},</math> and <math>\overline{MB}</math> are consecutive odd positive integers (in this order). What is the volume of pyramid <math>MACD?</math><br />
<br />
<math>\textbf{(A) }24\sqrt5 \qquad \textbf{(B) }60 \qquad \textbf{(C) }28\sqrt5\qquad \textbf{(D) }66 \qquad \textbf{(E) }8\sqrt{70}</math><br />
<br />
[[2021 AMC 12B Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
The figure is constructed from <math>11</math> line segments, each of which has length <math>2</math>. The area of pentagon <math>ABCDE</math> can be written is <math>\sqrt{m} + \sqrt{n}</math>, where <math>m</math> and <math>n</math> are positive integers. What is <math>m + n ?</math><br />
<asy> /* Made by samrocksnature */ pair A=(-2.4638,4.10658); pair B=(-4,2.6567453480756127); pair C=(-3.47132,0.6335248637894945); pair D=(-1.464483379039766,0.6335248637894945); pair E=(-0.956630463955801,2.6567453480756127); pair F=(-2,2); pair G=(-3,2); draw(A--B--C--D--E--A); draw(A--F--A--G); draw(B--F--C); draw(E--G--D); label("A",A,N); label("B",B,W); label("C",C,W); label("D",D,E); label("E",E,dir(0)); dot(A^^B^^C^^D^^E^^F^^G); </asy><br />
<math>\textbf{(A)} ~20 \qquad\textbf{(B)} ~21 \qquad\textbf{(C)} ~22 \qquad\textbf{(D)} ~23 \qquad\textbf{(E)} ~24</math><br />
<br />
[[2021 AMC 12B Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
Let <math>g(x)</math> be a polynomial with leading coefficient <math>1,</math> whose three roots are the reciprocals of the three roots of <math>f(x)=x^3+ax^2+bx+c,</math> where <math>1<a<b<c.</math> What is <math>g(1)</math> in terms of <math>a,b,</math> and <math>c?</math><br />
<br />
<math>\textbf{(A) }\frac{1+a+b+c}c \qquad \textbf{(B) }1+a+b+c \qquad \textbf{(C) }\frac{1+a+b+c}{c^2}\qquad \textbf{(D) }\frac{a+b+c}{c^2} \qquad \textbf{(E) }\frac{1+a+b+c}{a+b+c}</math><br />
<br />
[[2021 AMC 12B Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
Let <math>ABCD</math> be an isoceles trapezoid having parallel bases <math>\overline{AB}</math> and <math>\overline{CD}</math> with <math>AB>CD.</math> Line segments from a point inside <math>ABCD</math> to the vertices divide the trapezoid into four triangles whose areas are <math>2, 3, 4,</math> and <math>5</math> starting with the triangle with base <math>\overline{CD}</math> and moving clockwise as shown in the diagram below. What is the ratio <math>\frac{AB}{CD}?</math><br />
<asy><br />
unitsize(100);<br />
pair A=(-1, 0), B=(1, 0), C=(0.3, 0.9), D=(-0.3, 0.9), P=(0.2, 0.5), E=(0.1, 0.75), F=(0.4, 0.5), G=(0.15, 0.2), H=(-0.3, 0.5); <br />
draw(A--B--C--D--cycle, black); <br />
draw(A--P, black);<br />
draw(B--P, black);<br />
draw(C--P, black);<br />
draw(D--P, black);<br />
label("$A$",A,(-1,0));<br />
label("$B$",B,(1,0));<br />
label("$C$",C,(1,-0));<br />
label("$D$",D,(-1,0));<br />
label("$2$",E,(0,0));<br />
label("$3$",F,(0,0));<br />
label("$4$",G,(0,0));<br />
label("$5$",H,(0,0));<br />
dot(A^^B^^C^^D^^P);<br />
</asy><br />
<math>\textbf{(A)}\: 3\qquad\textbf{(B)}\: 2+\sqrt{2}\qquad\textbf{(C)}\: 1+\sqrt{6}\qquad\textbf{(D)}\: 2\sqrt{3}\qquad\textbf{(E)}\: 3\sqrt{2}</math><br />
<br />
[[2021 AMC 12B Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
Let <math>z</math> be a complex number satisfying <math>12|z|^2=2|z+2|^2+|z^2+1|^2+31.</math> What is the value of <math>z+\frac 6z?</math><br />
<br />
<math>\textbf{(A) }-2 \qquad \textbf{(B) }-1 \qquad \textbf{(C) }\frac12\qquad \textbf{(D) }1 \qquad \textbf{(E) }4</math><br />
<br />
[[2021 AMC 12B Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
Two fair dice, each with at least <math>6</math> faces are rolled. On each face of each dice is printed a distinct integer from <math>1</math> to the number of faces on that die, inclusive. The probability of rolling a sum if <math>7</math> is <math>\frac34</math> of the probability of rolling a sum of <math>10,</math> and the probability of rolling a sum of <math>12</math> is <math>\frac{1}{12}</math>. What is the least possible number of faces on the two dice combined?<br />
<br />
<math>\textbf{(A) }16 \qquad \textbf{(B) }17 \qquad \textbf{(C) }18\qquad \textbf{(D) }19 \qquad \textbf{(E) }20</math><br />
<br />
[[2021 AMC 12B Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
Let <math>Q(z)</math> and <math>R(z)</math> be the unique polynomials such that<cmath>z^{2021}+1=(z^2+z+1)Q(z)+R(z)</cmath>and the degree of <math>R</math> is less than <math>2.</math> What is <math>R(z)?</math><br />
<br />
<math>\textbf{(A) }-z \qquad \textbf{(B) }-1 \qquad \textbf{(C) }2021\qquad \textbf{(D) }z+1 \qquad \textbf{(E) }2z+1</math><br />
<br />
[[2021 AMC 12B Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
Let <math>S</math> be the sum of all positive real numbers <math>x</math> for which<cmath>x^{2^{\sqrt2}}=\sqrt2^{2^x}.</cmath>Which of the following statements is true?<br />
<br />
<math>\textbf{(A) }S<\sqrt2 \qquad \textbf{(B) }S=\sqrt2 \qquad \textbf{(C) }\sqrt2<S<2\qquad \textbf{(D) }2\le S<6 \qquad \textbf{(E) }S\ge 6</math><br />
<br />
[[2021 AMC 12B Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
Arjun and Beth play a game in which they take turns removing one brick or two adjacent bricks from one "wall" among a set of several walls of bricks, with gaps possibly creating new walls. The walls are one brick tall. For example, a set of walls of sizes <math>4</math> and <math>2</math> can be changed into any of the following by one move: <math>(3,2),(2,1,2),(4),(4,1),(2,2),</math> or <math>(1,1,2).</math> <br />
<br />
<asy><br />
unitsize(4mm);<br />
real[] boxes = {0,1,2,3,5,6,13,14,15,17,18,21,22,24,26,27,30,31,32,33};<br />
for(real i:boxes){<br />
draw(box((i,0),(i+1,3)));<br />
}<br />
draw((8,1.5)--(12,1.5),Arrow());<br />
defaultpen(fontsize(20pt));<br />
label(",",(20,0));<br />
label(",",(29,0));<br />
label(",...",(35.5,0));<br />
</asy><br />
<br />
Arjun plays first, and the player who removes the last brick wins. For which starting configuration is there a strategy that guarantees a win for Beth?<br />
<br />
<math>\textbf{(A) }(6,1,1) \qquad \textbf{(B) }(6,2,1) \qquad \textbf{(C) }(6,2,2)\qquad \textbf{(D) }(6,3,1) \qquad \textbf{(E) }(6,3,2)</math><br />
<br />
[[2021 AMC 12B Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
Three balls are randomly and independantly tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin <math>i</math> is <math>2^{-i}</math> for <math>i=1,2,3,....</math> More than one ball is allowed in each bin. The probability that the balls end up evenly spaced in distinct bins is <math>\frac pq,</math> where <math>p</math> and <math>q</math> are relatively prime positive integers. (For example, the balls are evenly spaced if they are tossed into bins <math>3,17,</math> and <math>10.</math>) What is <math>p+q?</math><br />
<br />
<math>\textbf{(A) }55 \qquad \textbf{(B) }56 \qquad \textbf{(C) }57\qquad \textbf{(D) }58 \qquad \textbf{(E) }59</math><br />
<br />
[[2021 AMC 12B Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
Let <math>ABCD</math> be a parallelogram with area <math>15</math>. Points <math>P</math> and <math>Q</math> are the projections of <math>A</math> and <math>C,</math> respectively, onto the line <math>BD;</math> and points <math>R</math> and <math>S</math> are the projections of <math>B</math> and <math>D,</math> respectively, onto the line <math>AC.</math> See the figure, which also shows the relative locations of these points.<br />
<br />
<asy><br />
size(350);<br />
defaultpen(linewidth(0.8)+fontsize(11));<br />
real theta = aTan(1.25/2);<br />
pair A = 2.5*dir(180+theta), B = (3.35,0), C = -A, D = -B, P = foot(A,B,D), Q = -P, R = foot(B,A,C), S = -R;<br />
draw(A--B--C--D--A^^B--D^^R--S^^rightanglemark(A,P,D,6)^^rightanglemark(C,Q,D,6));<br />
draw(B--R^^C--Q^^A--P^^D--S,linetype("4 4"));<br />
dot("$A$",A,dir(270));<br />
dot("$B$",B,E);<br />
dot("$C$",C,N);<br />
dot("$D$",D,W);<br />
dot("$P$",P,SE);<br />
dot("$Q$",Q,NE);<br />
dot("$R$",R,N);<br />
dot("$S$",S,dir(270));<br />
</asy><br />
<br />
Suppose <math>PQ=6</math> and <math>RS=8,</math> and let <math>d</math> denote the length of <math>\overline{BD},</math> the longer diagonal of <math>ABCD.</math> Then <math>d^2</math> can be written in the form <math>m+n\sqrt p,</math> where <math>m,n,</math> and <math>p</math> are positive integers and <math>p</math> is not divisible by the square of any prime. What is <math>m+n+p?</math><br />
<br />
<math>\textbf{(A) }81 \qquad \textbf{(B) }89 \qquad \textbf{(C) }97\qquad \textbf{(D) }105 \qquad \textbf{(E) }113</math><br />
<br />
[[2021 AMC 12B Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
Let <math>S</math> be the set of lattice points in the coordinate plane, both of whose coordinates are integers between <math>1</math> and <math>30,</math> inclusive. Exactly <math>300</math> points in <math>S</math> lie on or below a line with equation <math>y=mx.</math> The possible values of <math>m</math> lie in an interval of length <math>\frac ab,</math> where <math>a</math> and <math>b</math> are relatively prime positive integers. What is <math>a+b?</math><br />
<br />
<math>\textbf{(A) }31 \qquad \textbf{(B) }47 \qquad \textbf{(C) }62\qquad \textbf{(D) }72 \qquad \textbf{(E) }85</math><br />
<br />
[[2021 AMC 12B Problems/Problem 25|Solution]]<br />
<br />
<br />
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 900 lattice points. I estimate that the slope, m, 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 600 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 are is above the line. <br />
<br />
I find that the number of coordinates with <math>x=1</math> above the line is 30, 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 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 <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 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. <br />
<br />
To find the upper bound, notice that each point with an integer-valued x-coordinate is either <math>\frac{1}{3}</math> or <math>\frac{2}{3}</math> 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 <math>x=28, 29, 30</math> which the line <math>y=\frac{2}{3}x</math> intersects at <math>y= \frac{56}{3}, \frac{58}{3}, 20</math>. 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 slope of the line which goes through both the origin and (28,19) is <math>y=\frac{19}{28}x</math>. This gives an upper bound of <math>m=\frac{19}{28}</math>. <br />
<br />
Taking the upper bound of m and subtracting the lower bound yields <math>\frac{19}{28}-\frac{2}{3}=\frac{1}{84}</math>. This is answer E.<br />
<br />
==See also==<br />
{{AMC12 box|year=2021|ab=B|before=[[2021 AMC 12A Problems]]|after=[[2022 AMC 12A Problems]]}}<br />
<br />
[[Category:AMC 12 Problems]]<br />
{{MAA Notice}}</div>Theajlhttps://artofproblemsolving.com/wiki/index.php?title=2021_AMC_12B_Problems&diff=1457142021 AMC 12B Problems2021-02-12T03:53:26Z<p>Theajl: /* Problem 25 */</p>
<hr />
<div>{{AMC12 Problems|year=2021|ab=B}}<br />
==Problem 1==<br />
How many integer values of <math>x</math> satisfy <math>|x|<3\pi?</math><br />
<br />
<math>\textbf{(A) }9 \qquad \textbf{(B) }10 \qquad \textbf{(C) }18 \qquad \textbf{(D) }19 \qquad \textbf{(E) }20</math><br />
<br />
[[2021 AMC 12B Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
At a math contest, <math>57</math> students are wearing blue shirts, and another <math>75</math> students are wearing yellow shirts. The <math>132</math> students are assigned into <math>66</math> points. In exactly <math>23</math> of these pairs, both students are wearing blue shirts. In how many pairs are both studets wearing yellow shirts?<br />
<br />
<math>\textbf{(A) }23 \qquad \textbf{(B) }32 \qquad \textbf{(C) }37 \qquad \textbf{(D) }41 \qquad \textbf{(E) }64</math><br />
<br />
[[2021 AMC 12B Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
Suppose<cmath>2+\frac{1}{1+\frac{1}{2+\frac{2}{3+x}}}=\frac{144}{53}.</cmath>What is the value of <math>x?</math><br />
<br />
<math>\textbf{(A) }\frac34 \qquad \textbf{(B) }\frac78 \qquad \textbf{(C) }\frac{14}{15} \qquad \textbf{(D) }\frac{37}{38} \qquad \textbf{(E) }\frac{52}{53}</math><br />
<br />
[[2021 AMC 12B Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
Ms. Blackwell gives an exam to two classes. The mean of the scores of the students in the morning class is <math>84</math>, and the afternoon class's mean score is <math>70</math>. The ratio of the number of students in the morning clas to the number of students in the afternoon class is <math>\frac34</math>. What is the mean of the score of all the students?<br />
<br />
<math>\textbf{(A) }74 \qquad \textbf{(B) }75 \qquad \textbf{(C) }76 \qquad \textbf{(D) }77 \qquad \textbf{(E) }78</math><br />
<br />
[[2021 AMC 12B Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
The point <math>P(a,b)</math> in the <math>xy</math>-plane is first rotated counterclockwise by <math>90^\circ</math> around the point <math>(1,5)</math> and then reflected about the line <math>y=-x</math>. The image of <math>P</math> after these two transformations is at <math>(-6,3)</math>. What is <math>b-a?</math><br />
<br />
<math>\textbf{(A) }1 \qquad \textbf{(B) }3 \qquad \textbf{(C) }5 \qquad \textbf{(D) }7 \qquad \textbf{(E) }9</math><br />
<br />
[[2021 AMC 12B Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
An inverted cone with base radius <math>12 \text{cm}</math> and height <math>18\text{cm}</math> is full of water. The water is poured into a tall cylinder whose horizontal base has a radius of <math>24\text{cm}</math>. What is the height in centimeters of the water in the cylinder?<br />
<br />
<math>\textbf{(A) }1.5 \qquad \textbf{(B) }3 \qquad \textbf{(C) }4 \qquad \textbf{(D) }4.5 \qquad \textbf{(E) }6</math><br />
<br />
[[2021 AMC 12B Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
Let <math>N=34\cdot34\cdot63\cdot270.</math> What is the ratio of the sum of the odd divisors of <math>N</math> to the sum of the even divisors of <math>N?</math><br />
<br />
<math>\textbf{(A) }1:16 \qquad \textbf{(B) }1:15 \qquad \textbf{(C) }1:14 \qquad \textbf{(D) }1:8 \qquad \textbf{(E) }1:3</math><br />
<br />
[[2021 AMC 12B Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
Three equally spaced parallel lines intersect a circle, creating three chords of lengths <math>38,38,</math> and <math>34</math>. What is the distance between two adjacent parallel lines?<br />
<br />
<math>\textbf{(A) }5\frac12 \qquad \textbf{(B) }6 \qquad \textbf{(C) }6\frac12 \qquad \textbf{(D) }7 \qquad \textbf{(E) }7\frac12</math><br />
<br />
[[2021 AMC 12B Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
What is the value of<cmath>\frac{\log_2 80}{\log_{40}2}-\frac{\log_2 160}{\log_{20}2}?</cmath><br />
<math>\textbf{(A) }0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }\frac54 \qquad \textbf{(D) }2 \qquad \textbf{(E) }\log_2 5</math><br />
<br />
[[2021 AMC 12B Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
Two distinct numbers are selected from the set <math>\{1,2,3,4,\dots,36,37\}</math> so that the sum of the remaining <math>35</math> numbers is the product of these two numbers. What is the difference of these two numbers?<br />
<br />
<math>\textbf{(A) }5 \qquad \textbf{(B) }7 \qquad \textbf{(C) }8\qquad \textbf{(D) }9 \qquad \textbf{(E) }10</math><br />
<br />
[[2021 AMC 12B Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
Triangle <math>ABC</math> has <math>AB=13,BC=14</math> and <math>AC=15</math>. Let <math>P</math> be the point on <math>\overline{AC}</math> such that <math>PC=10</math>. There are exactly two points <math>D</math> and <math>E</math> on line <math>BP</math> such that quadrilaterals <math>ABCD</math> and <math>ABCE</math> are trapezoids. What is the distance <math>DE?</math><br />
<br />
<math>\textbf{(A) }\frac{42}5 \qquad \textbf{(B) }6\sqrt2 \qquad \textbf{(C) }\frac{84}5\qquad \textbf{(D) }12\sqrt2 \qquad \textbf{(E) }18</math><br />
<br />
[[2021 AMC 12B Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
Suppose that <math>S</math> is a finite set of positive integers. If the greatest integer in <math>S</math> is removed from <math>S</math>, then the average value (arithmetic mean) of the integers remaining is <math>32</math>. If the least integer in <math>S</math> is also removed, then the average value of the integers remaining is <math>35</math>. If the great integer is then returned to the set, the average value of the integers rises to <math>40.</math> The greatest integer in the original set <math>S</math> is <math>72</math> greater than the least integer in <math>S</math>. What is the average value of all the integers in the set <math>S?</math><br />
<br />
<math>\textbf{(A) }36.2 \qquad \textbf{(B) }36.4 \qquad \textbf{(C) }36.6\qquad \textbf{(D) }36.8 \qquad \textbf{(E) }37</math><br />
<br />
[[2021 AMC 12B Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
How many values of <math>\theta</math> in the interval <math>0<\theta\le 2\pi</math> satisfy<cmath>1-3\sin\theta+5\cos3\theta?</cmath><br />
<math>\textbf{(A) }2 \qquad \textbf{(B) }4 \qquad \textbf{(C) }5\qquad \textbf{(D) }6 \qquad \textbf{(E) }8</math><br />
<br />
[[2021 AMC 12B Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
Let <math>ABCD</math> be a rectangle and let <math>\overline{DM}</math> be a segment perpendicular to the plane of <math>ABCD</math>. Suppose that <math>\overline{DM}</math> has integer length, and the lengths of <math>\overline{MA},\overline{MC},</math> and <math>\overline{MB}</math> are consecutive odd positive integers (in this order). What is the volume of pyramid <math>MACD?</math><br />
<br />
<math>\textbf{(A) }24\sqrt5 \qquad \textbf{(B) }60 \qquad \textbf{(C) }28\sqrt5\qquad \textbf{(D) }66 \qquad \textbf{(E) }8\sqrt{70}</math><br />
<br />
[[2021 AMC 12B Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
The figure is constructed from <math>11</math> line segments, each of which has length <math>2</math>. The area of pentagon <math>ABCDE</math> can be written is <math>\sqrt{m} + \sqrt{n}</math>, where <math>m</math> and <math>n</math> are positive integers. What is <math>m + n ?</math><br />
<asy> /* Made by samrocksnature */ pair A=(-2.4638,4.10658); pair B=(-4,2.6567453480756127); pair C=(-3.47132,0.6335248637894945); pair D=(-1.464483379039766,0.6335248637894945); pair E=(-0.956630463955801,2.6567453480756127); pair F=(-2,2); pair G=(-3,2); draw(A--B--C--D--E--A); draw(A--F--A--G); draw(B--F--C); draw(E--G--D); label("A",A,N); label("B",B,W); label("C",C,W); label("D",D,E); label("E",E,dir(0)); dot(A^^B^^C^^D^^E^^F^^G); </asy><br />
<math>\textbf{(A)} ~20 \qquad\textbf{(B)} ~21 \qquad\textbf{(C)} ~22 \qquad\textbf{(D)} ~23 \qquad\textbf{(E)} ~24</math><br />
<br />
[[2021 AMC 12B Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
Let <math>g(x)</math> be a polynomial with leading coefficient <math>1,</math> whose three roots are the reciprocals of the three roots of <math>f(x)=x^3+ax^2+bx+c,</math> where <math>1<a<b<c.</math> What is <math>g(1)</math> in terms of <math>a,b,</math> and <math>c?</math><br />
<br />
<math>\textbf{(A) }\frac{1+a+b+c}c \qquad \textbf{(B) }1+a+b+c \qquad \textbf{(C) }\frac{1+a+b+c}{c^2}\qquad \textbf{(D) }\frac{a+b+c}{c^2} \qquad \textbf{(E) }\frac{1+a+b+c}{a+b+c}</math><br />
<br />
[[2021 AMC 12B Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
Let <math>ABCD</math> be an isoceles trapezoid having parallel bases <math>\overline{AB}</math> and <math>\overline{CD}</math> with <math>AB>CD.</math> Line segments from a point inside <math>ABCD</math> to the vertices divide the trapezoid into four triangles whose areas are <math>2, 3, 4,</math> and <math>5</math> starting with the triangle with base <math>\overline{CD}</math> and moving clockwise as shown in the diagram below. What is the ratio <math>\frac{AB}{CD}?</math><br />
<asy><br />
unitsize(100);<br />
pair A=(-1, 0), B=(1, 0), C=(0.3, 0.9), D=(-0.3, 0.9), P=(0.2, 0.5), E=(0.1, 0.75), F=(0.4, 0.5), G=(0.15, 0.2), H=(-0.3, 0.5); <br />
draw(A--B--C--D--cycle, black); <br />
draw(A--P, black);<br />
draw(B--P, black);<br />
draw(C--P, black);<br />
draw(D--P, black);<br />
label("$A$",A,(-1,0));<br />
label("$B$",B,(1,0));<br />
label("$C$",C,(1,-0));<br />
label("$D$",D,(-1,0));<br />
label("$2$",E,(0,0));<br />
label("$3$",F,(0,0));<br />
label("$4$",G,(0,0));<br />
label("$5$",H,(0,0));<br />
dot(A^^B^^C^^D^^P);<br />
</asy><br />
<math>\textbf{(A)}\: 3\qquad\textbf{(B)}\: 2+\sqrt{2}\qquad\textbf{(C)}\: 1+\sqrt{6}\qquad\textbf{(D)}\: 2\sqrt{3}\qquad\textbf{(E)}\: 3\sqrt{2}</math><br />
<br />
[[2021 AMC 12B Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
Let <math>z</math> be a complex number satisfying <math>12|z|^2=2|z+2|^2+|z^2+1|^2+31.</math> What is the value of <math>z+\frac 6z?</math><br />
<br />
<math>\textbf{(A) }-2 \qquad \textbf{(B) }-1 \qquad \textbf{(C) }\frac12\qquad \textbf{(D) }1 \qquad \textbf{(E) }4</math><br />
<br />
[[2021 AMC 12B Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
Two fair dice, each with at least <math>6</math> faces are rolled. On each face of each dice is printed a distinct integer from <math>1</math> to the number of faces on that die, inclusive. The probability of rolling a sum if <math>7</math> is <math>\frac34</math> of the probability of rolling a sum of <math>10,</math> and the probability of rolling a sum of <math>12</math> is <math>\frac{1}{12}</math>. What is the least possible number of faces on the two dice combined?<br />
<br />
<math>\textbf{(A) }16 \qquad \textbf{(B) }17 \qquad \textbf{(C) }18\qquad \textbf{(D) }19 \qquad \textbf{(E) }20</math><br />
<br />
[[2021 AMC 12B Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
Let <math>Q(z)</math> and <math>R(z)</math> be the unique polynomials such that<cmath>z^{2021}+1=(z^2+z+1)Q(z)+R(z)</cmath>and the degree of <math>R</math> is less than <math>2.</math> What is <math>R(z)?</math><br />
<br />
<math>\textbf{(A) }-z \qquad \textbf{(B) }-1 \qquad \textbf{(C) }2021\qquad \textbf{(D) }z+1 \qquad \textbf{(E) }2z+1</math><br />
<br />
[[2021 AMC 12B Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
Let <math>S</math> be the sum of all positive real numbers <math>x</math> for which<cmath>x^{2^{\sqrt2}}=\sqrt2^{2^x}.</cmath>Which of the following statements is true?<br />
<br />
<math>\textbf{(A) }S<\sqrt2 \qquad \textbf{(B) }S=\sqrt2 \qquad \textbf{(C) }\sqrt2<S<2\qquad \textbf{(D) }2\le S<6 \qquad \textbf{(E) }S\ge 6</math><br />
<br />
[[2021 AMC 12B Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
Arjun and Beth play a game in which they take turns removing one brick or two adjacent bricks from one "wall" among a set of several walls of bricks, with gaps possibly creating new walls. The walls are one brick tall. For example, a set of walls of sizes <math>4</math> and <math>2</math> can be changed into any of the following by one move: <math>(3,2),(2,1,2),(4),(4,1),(2,2),</math> or <math>(1,1,2).</math> <br />
<br />
<asy><br />
unitsize(4mm);<br />
real[] boxes = {0,1,2,3,5,6,13,14,15,17,18,21,22,24,26,27,30,31,32,33};<br />
for(real i:boxes){<br />
draw(box((i,0),(i+1,3)));<br />
}<br />
draw((8,1.5)--(12,1.5),Arrow());<br />
defaultpen(fontsize(20pt));<br />
label(",",(20,0));<br />
label(",",(29,0));<br />
label(",...",(35.5,0));<br />
</asy><br />
<br />
Arjun plays first, and the player who removes the last brick wins. For which starting configuration is there a strategy that guarantees a win for Beth?<br />
<br />
<math>\textbf{(A) }(6,1,1) \qquad \textbf{(B) }(6,2,1) \qquad \textbf{(C) }(6,2,2)\qquad \textbf{(D) }(6,3,1) \qquad \textbf{(E) }(6,3,2)</math><br />
<br />
[[2021 AMC 12B Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
Three balls are randomly and independantly tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin <math>i</math> is <math>2^{-i}</math> for <math>i=1,2,3,....</math> More than one ball is allowed in each bin. The probability that the balls end up evenly spaced in distinct bins is <math>\frac pq,</math> where <math>p</math> and <math>q</math> are relatively prime positive integers. (For example, the balls are evenly spaced if they are tossed into bins <math>3,17,</math> and <math>10.</math>) What is <math>p+q?</math><br />
<br />
<math>\textbf{(A) }55 \qquad \textbf{(B) }56 \qquad \textbf{(C) }57\qquad \textbf{(D) }58 \qquad \textbf{(E) }59</math><br />
<br />
[[2021 AMC 12B Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
Let <math>ABCD</math> be a parallelogram with area <math>15</math>. Points <math>P</math> and <math>Q</math> are the projections of <math>A</math> and <math>C,</math> respectively, onto the line <math>BD;</math> and points <math>R</math> and <math>S</math> are the projections of <math>B</math> and <math>D,</math> respectively, onto the line <math>AC.</math> See the figure, which also shows the relative locations of these points.<br />
<br />
<asy><br />
size(350);<br />
defaultpen(linewidth(0.8)+fontsize(11));<br />
real theta = aTan(1.25/2);<br />
pair A = 2.5*dir(180+theta), B = (3.35,0), C = -A, D = -B, P = foot(A,B,D), Q = -P, R = foot(B,A,C), S = -R;<br />
draw(A--B--C--D--A^^B--D^^R--S^^rightanglemark(A,P,D,6)^^rightanglemark(C,Q,D,6));<br />
draw(B--R^^C--Q^^A--P^^D--S,linetype("4 4"));<br />
dot("$A$",A,dir(270));<br />
dot("$B$",B,E);<br />
dot("$C$",C,N);<br />
dot("$D$",D,W);<br />
dot("$P$",P,SE);<br />
dot("$Q$",Q,NE);<br />
dot("$R$",R,N);<br />
dot("$S$",S,dir(270));<br />
</asy><br />
<br />
Suppose <math>PQ=6</math> and <math>RS=8,</math> and let <math>d</math> denote the length of <math>\overline{BD},</math> the longer diagonal of <math>ABCD.</math> Then <math>d^2</math> can be written in the form <math>m+n\sqrt p,</math> where <math>m,n,</math> and <math>p</math> are positive integers and <math>p</math> is not divisible by the square of any prime. What is <math>m+n+p?</math><br />
<br />
<math>\textbf{(A) }81 \qquad \textbf{(B) }89 \qquad \textbf{(C) }97\qquad \textbf{(D) }105 \qquad \textbf{(E) }113</math><br />
<br />
[[2021 AMC 12B Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
Let <math>S</math> be the set of lattice points in the coordinate plane, both of whose coordinates are integers between <math>1</math> and <math>30,</math> inclusive. Exactly <math>300</math> points in <math>S</math> lie on or below a line with equation <math>y=mx.</math> The possible values of <math>m</math> lie in an interval of length <math>\frac ab,</math> where <math>a</math> and <math>b</math> are relatively prime positive integers. What is <math>a+b?</math><br />
<br />
<math>\textbf{(A) }31 \qquad \textbf{(B) }47 \qquad \textbf{(C) }62\qquad \textbf{(D) }72 \qquad \textbf{(E) }85</math><br />
<br />
[[2021 AMC 12B Problems/Problem 25|Solution]]<br />
<br />
==See also==<br />
{{AMC12 box|year=2021|ab=B|before=[[2021 AMC 12A Problems]]|after=[[2022 AMC 12A Problems]]}}<br />
<br />
[[Category:AMC 12 Problems]]<br />
{{MAA Notice}}</div>Theajlhttps://artofproblemsolving.com/wiki/index.php?title=2021_AMC_12B_Problems&diff=1457132021 AMC 12B Problems2021-02-12T03:52:34Z<p>Theajl: /* Problem 25 */</p>
<hr />
<div>{{AMC12 Problems|year=2021|ab=B}}<br />
==Problem 1==<br />
How many integer values of <math>x</math> satisfy <math>|x|<3\pi?</math><br />
<br />
<math>\textbf{(A) }9 \qquad \textbf{(B) }10 \qquad \textbf{(C) }18 \qquad \textbf{(D) }19 \qquad \textbf{(E) }20</math><br />
<br />
[[2021 AMC 12B Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
At a math contest, <math>57</math> students are wearing blue shirts, and another <math>75</math> students are wearing yellow shirts. The <math>132</math> students are assigned into <math>66</math> points. In exactly <math>23</math> of these pairs, both students are wearing blue shirts. In how many pairs are both studets wearing yellow shirts?<br />
<br />
<math>\textbf{(A) }23 \qquad \textbf{(B) }32 \qquad \textbf{(C) }37 \qquad \textbf{(D) }41 \qquad \textbf{(E) }64</math><br />
<br />
[[2021 AMC 12B Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
Suppose<cmath>2+\frac{1}{1+\frac{1}{2+\frac{2}{3+x}}}=\frac{144}{53}.</cmath>What is the value of <math>x?</math><br />
<br />
<math>\textbf{(A) }\frac34 \qquad \textbf{(B) }\frac78 \qquad \textbf{(C) }\frac{14}{15} \qquad \textbf{(D) }\frac{37}{38} \qquad \textbf{(E) }\frac{52}{53}</math><br />
<br />
[[2021 AMC 12B Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
Ms. Blackwell gives an exam to two classes. The mean of the scores of the students in the morning class is <math>84</math>, and the afternoon class's mean score is <math>70</math>. The ratio of the number of students in the morning clas to the number of students in the afternoon class is <math>\frac34</math>. What is the mean of the score of all the students?<br />
<br />
<math>\textbf{(A) }74 \qquad \textbf{(B) }75 \qquad \textbf{(C) }76 \qquad \textbf{(D) }77 \qquad \textbf{(E) }78</math><br />
<br />
[[2021 AMC 12B Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
The point <math>P(a,b)</math> in the <math>xy</math>-plane is first rotated counterclockwise by <math>90^\circ</math> around the point <math>(1,5)</math> and then reflected about the line <math>y=-x</math>. The image of <math>P</math> after these two transformations is at <math>(-6,3)</math>. What is <math>b-a?</math><br />
<br />
<math>\textbf{(A) }1 \qquad \textbf{(B) }3 \qquad \textbf{(C) }5 \qquad \textbf{(D) }7 \qquad \textbf{(E) }9</math><br />
<br />
[[2021 AMC 12B Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
An inverted cone with base radius <math>12 \text{cm}</math> and height <math>18\text{cm}</math> is full of water. The water is poured into a tall cylinder whose horizontal base has a radius of <math>24\text{cm}</math>. What is the height in centimeters of the water in the cylinder?<br />
<br />
<math>\textbf{(A) }1.5 \qquad \textbf{(B) }3 \qquad \textbf{(C) }4 \qquad \textbf{(D) }4.5 \qquad \textbf{(E) }6</math><br />
<br />
[[2021 AMC 12B Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
Let <math>N=34\cdot34\cdot63\cdot270.</math> What is the ratio of the sum of the odd divisors of <math>N</math> to the sum of the even divisors of <math>N?</math><br />
<br />
<math>\textbf{(A) }1:16 \qquad \textbf{(B) }1:15 \qquad \textbf{(C) }1:14 \qquad \textbf{(D) }1:8 \qquad \textbf{(E) }1:3</math><br />
<br />
[[2021 AMC 12B Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
Three equally spaced parallel lines intersect a circle, creating three chords of lengths <math>38,38,</math> and <math>34</math>. What is the distance between two adjacent parallel lines?<br />
<br />
<math>\textbf{(A) }5\frac12 \qquad \textbf{(B) }6 \qquad \textbf{(C) }6\frac12 \qquad \textbf{(D) }7 \qquad \textbf{(E) }7\frac12</math><br />
<br />
[[2021 AMC 12B Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
What is the value of<cmath>\frac{\log_2 80}{\log_{40}2}-\frac{\log_2 160}{\log_{20}2}?</cmath><br />
<math>\textbf{(A) }0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }\frac54 \qquad \textbf{(D) }2 \qquad \textbf{(E) }\log_2 5</math><br />
<br />
[[2021 AMC 12B Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
Two distinct numbers are selected from the set <math>\{1,2,3,4,\dots,36,37\}</math> so that the sum of the remaining <math>35</math> numbers is the product of these two numbers. What is the difference of these two numbers?<br />
<br />
<math>\textbf{(A) }5 \qquad \textbf{(B) }7 \qquad \textbf{(C) }8\qquad \textbf{(D) }9 \qquad \textbf{(E) }10</math><br />
<br />
[[2021 AMC 12B Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
Triangle <math>ABC</math> has <math>AB=13,BC=14</math> and <math>AC=15</math>. Let <math>P</math> be the point on <math>\overline{AC}</math> such that <math>PC=10</math>. There are exactly two points <math>D</math> and <math>E</math> on line <math>BP</math> such that quadrilaterals <math>ABCD</math> and <math>ABCE</math> are trapezoids. What is the distance <math>DE?</math><br />
<br />
<math>\textbf{(A) }\frac{42}5 \qquad \textbf{(B) }6\sqrt2 \qquad \textbf{(C) }\frac{84}5\qquad \textbf{(D) }12\sqrt2 \qquad \textbf{(E) }18</math><br />
<br />
[[2021 AMC 12B Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
Suppose that <math>S</math> is a finite set of positive integers. If the greatest integer in <math>S</math> is removed from <math>S</math>, then the average value (arithmetic mean) of the integers remaining is <math>32</math>. If the least integer in <math>S</math> is also removed, then the average value of the integers remaining is <math>35</math>. If the great integer is then returned to the set, the average value of the integers rises to <math>40.</math> The greatest integer in the original set <math>S</math> is <math>72</math> greater than the least integer in <math>S</math>. What is the average value of all the integers in the set <math>S?</math><br />
<br />
<math>\textbf{(A) }36.2 \qquad \textbf{(B) }36.4 \qquad \textbf{(C) }36.6\qquad \textbf{(D) }36.8 \qquad \textbf{(E) }37</math><br />
<br />
[[2021 AMC 12B Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
How many values of <math>\theta</math> in the interval <math>0<\theta\le 2\pi</math> satisfy<cmath>1-3\sin\theta+5\cos3\theta?</cmath><br />
<math>\textbf{(A) }2 \qquad \textbf{(B) }4 \qquad \textbf{(C) }5\qquad \textbf{(D) }6 \qquad \textbf{(E) }8</math><br />
<br />
[[2021 AMC 12B Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
Let <math>ABCD</math> be a rectangle and let <math>\overline{DM}</math> be a segment perpendicular to the plane of <math>ABCD</math>. Suppose that <math>\overline{DM}</math> has integer length, and the lengths of <math>\overline{MA},\overline{MC},</math> and <math>\overline{MB}</math> are consecutive odd positive integers (in this order). What is the volume of pyramid <math>MACD?</math><br />
<br />
<math>\textbf{(A) }24\sqrt5 \qquad \textbf{(B) }60 \qquad \textbf{(C) }28\sqrt5\qquad \textbf{(D) }66 \qquad \textbf{(E) }8\sqrt{70}</math><br />
<br />
[[2021 AMC 12B Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
The figure is constructed from <math>11</math> line segments, each of which has length <math>2</math>. The area of pentagon <math>ABCDE</math> can be written is <math>\sqrt{m} + \sqrt{n}</math>, where <math>m</math> and <math>n</math> are positive integers. What is <math>m + n ?</math><br />
<asy> /* Made by samrocksnature */ pair A=(-2.4638,4.10658); pair B=(-4,2.6567453480756127); pair C=(-3.47132,0.6335248637894945); pair D=(-1.464483379039766,0.6335248637894945); pair E=(-0.956630463955801,2.6567453480756127); pair F=(-2,2); pair G=(-3,2); draw(A--B--C--D--E--A); draw(A--F--A--G); draw(B--F--C); draw(E--G--D); label("A",A,N); label("B",B,W); label("C",C,W); label("D",D,E); label("E",E,dir(0)); dot(A^^B^^C^^D^^E^^F^^G); </asy><br />
<math>\textbf{(A)} ~20 \qquad\textbf{(B)} ~21 \qquad\textbf{(C)} ~22 \qquad\textbf{(D)} ~23 \qquad\textbf{(E)} ~24</math><br />
<br />
[[2021 AMC 12B Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
Let <math>g(x)</math> be a polynomial with leading coefficient <math>1,</math> whose three roots are the reciprocals of the three roots of <math>f(x)=x^3+ax^2+bx+c,</math> where <math>1<a<b<c.</math> What is <math>g(1)</math> in terms of <math>a,b,</math> and <math>c?</math><br />
<br />
<math>\textbf{(A) }\frac{1+a+b+c}c \qquad \textbf{(B) }1+a+b+c \qquad \textbf{(C) }\frac{1+a+b+c}{c^2}\qquad \textbf{(D) }\frac{a+b+c}{c^2} \qquad \textbf{(E) }\frac{1+a+b+c}{a+b+c}</math><br />
<br />
[[2021 AMC 12B Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
Let <math>ABCD</math> be an isoceles trapezoid having parallel bases <math>\overline{AB}</math> and <math>\overline{CD}</math> with <math>AB>CD.</math> Line segments from a point inside <math>ABCD</math> to the vertices divide the trapezoid into four triangles whose areas are <math>2, 3, 4,</math> and <math>5</math> starting with the triangle with base <math>\overline{CD}</math> and moving clockwise as shown in the diagram below. What is the ratio <math>\frac{AB}{CD}?</math><br />
<asy><br />
unitsize(100);<br />
pair A=(-1, 0), B=(1, 0), C=(0.3, 0.9), D=(-0.3, 0.9), P=(0.2, 0.5), E=(0.1, 0.75), F=(0.4, 0.5), G=(0.15, 0.2), H=(-0.3, 0.5); <br />
draw(A--B--C--D--cycle, black); <br />
draw(A--P, black);<br />
draw(B--P, black);<br />
draw(C--P, black);<br />
draw(D--P, black);<br />
label("$A$",A,(-1,0));<br />
label("$B$",B,(1,0));<br />
label("$C$",C,(1,-0));<br />
label("$D$",D,(-1,0));<br />
label("$2$",E,(0,0));<br />
label("$3$",F,(0,0));<br />
label("$4$",G,(0,0));<br />
label("$5$",H,(0,0));<br />
dot(A^^B^^C^^D^^P);<br />
</asy><br />
<math>\textbf{(A)}\: 3\qquad\textbf{(B)}\: 2+\sqrt{2}\qquad\textbf{(C)}\: 1+\sqrt{6}\qquad\textbf{(D)}\: 2\sqrt{3}\qquad\textbf{(E)}\: 3\sqrt{2}</math><br />
<br />
[[2021 AMC 12B Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
Let <math>z</math> be a complex number satisfying <math>12|z|^2=2|z+2|^2+|z^2+1|^2+31.</math> What is the value of <math>z+\frac 6z?</math><br />
<br />
<math>\textbf{(A) }-2 \qquad \textbf{(B) }-1 \qquad \textbf{(C) }\frac12\qquad \textbf{(D) }1 \qquad \textbf{(E) }4</math><br />
<br />
[[2021 AMC 12B Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
Two fair dice, each with at least <math>6</math> faces are rolled. On each face of each dice is printed a distinct integer from <math>1</math> to the number of faces on that die, inclusive. The probability of rolling a sum if <math>7</math> is <math>\frac34</math> of the probability of rolling a sum of <math>10,</math> and the probability of rolling a sum of <math>12</math> is <math>\frac{1}{12}</math>. What is the least possible number of faces on the two dice combined?<br />
<br />
<math>\textbf{(A) }16 \qquad \textbf{(B) }17 \qquad \textbf{(C) }18\qquad \textbf{(D) }19 \qquad \textbf{(E) }20</math><br />
<br />
[[2021 AMC 12B Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
Let <math>Q(z)</math> and <math>R(z)</math> be the unique polynomials such that<cmath>z^{2021}+1=(z^2+z+1)Q(z)+R(z)</cmath>and the degree of <math>R</math> is less than <math>2.</math> What is <math>R(z)?</math><br />
<br />
<math>\textbf{(A) }-z \qquad \textbf{(B) }-1 \qquad \textbf{(C) }2021\qquad \textbf{(D) }z+1 \qquad \textbf{(E) }2z+1</math><br />
<br />
[[2021 AMC 12B Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
Let <math>S</math> be the sum of all positive real numbers <math>x</math> for which<cmath>x^{2^{\sqrt2}}=\sqrt2^{2^x}.</cmath>Which of the following statements is true?<br />
<br />
<math>\textbf{(A) }S<\sqrt2 \qquad \textbf{(B) }S=\sqrt2 \qquad \textbf{(C) }\sqrt2<S<2\qquad \textbf{(D) }2\le S<6 \qquad \textbf{(E) }S\ge 6</math><br />
<br />
[[2021 AMC 12B Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
Arjun and Beth play a game in which they take turns removing one brick or two adjacent bricks from one "wall" among a set of several walls of bricks, with gaps possibly creating new walls. The walls are one brick tall. For example, a set of walls of sizes <math>4</math> and <math>2</math> can be changed into any of the following by one move: <math>(3,2),(2,1,2),(4),(4,1),(2,2),</math> or <math>(1,1,2).</math> <br />
<br />
<asy><br />
unitsize(4mm);<br />
real[] boxes = {0,1,2,3,5,6,13,14,15,17,18,21,22,24,26,27,30,31,32,33};<br />
for(real i:boxes){<br />
draw(box((i,0),(i+1,3)));<br />
}<br />
draw((8,1.5)--(12,1.5),Arrow());<br />
defaultpen(fontsize(20pt));<br />
label(",",(20,0));<br />
label(",",(29,0));<br />
label(",...",(35.5,0));<br />
</asy><br />
<br />
Arjun plays first, and the player who removes the last brick wins. For which starting configuration is there a strategy that guarantees a win for Beth?<br />
<br />
<math>\textbf{(A) }(6,1,1) \qquad \textbf{(B) }(6,2,1) \qquad \textbf{(C) }(6,2,2)\qquad \textbf{(D) }(6,3,1) \qquad \textbf{(E) }(6,3,2)</math><br />
<br />
[[2021 AMC 12B Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
Three balls are randomly and independantly tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin <math>i</math> is <math>2^{-i}</math> for <math>i=1,2,3,....</math> More than one ball is allowed in each bin. The probability that the balls end up evenly spaced in distinct bins is <math>\frac pq,</math> where <math>p</math> and <math>q</math> are relatively prime positive integers. (For example, the balls are evenly spaced if they are tossed into bins <math>3,17,</math> and <math>10.</math>) What is <math>p+q?</math><br />
<br />
<math>\textbf{(A) }55 \qquad \textbf{(B) }56 \qquad \textbf{(C) }57\qquad \textbf{(D) }58 \qquad \textbf{(E) }59</math><br />
<br />
[[2021 AMC 12B Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
Let <math>ABCD</math> be a parallelogram with area <math>15</math>. Points <math>P</math> and <math>Q</math> are the projections of <math>A</math> and <math>C,</math> respectively, onto the line <math>BD;</math> and points <math>R</math> and <math>S</math> are the projections of <math>B</math> and <math>D,</math> respectively, onto the line <math>AC.</math> See the figure, which also shows the relative locations of these points.<br />
<br />
<asy><br />
size(350);<br />
defaultpen(linewidth(0.8)+fontsize(11));<br />
real theta = aTan(1.25/2);<br />
pair A = 2.5*dir(180+theta), B = (3.35,0), C = -A, D = -B, P = foot(A,B,D), Q = -P, R = foot(B,A,C), S = -R;<br />
draw(A--B--C--D--A^^B--D^^R--S^^rightanglemark(A,P,D,6)^^rightanglemark(C,Q,D,6));<br />
draw(B--R^^C--Q^^A--P^^D--S,linetype("4 4"));<br />
dot("$A$",A,dir(270));<br />
dot("$B$",B,E);<br />
dot("$C$",C,N);<br />
dot("$D$",D,W);<br />
dot("$P$",P,SE);<br />
dot("$Q$",Q,NE);<br />
dot("$R$",R,N);<br />
dot("$S$",S,dir(270));<br />
</asy><br />
<br />
Suppose <math>PQ=6</math> and <math>RS=8,</math> and let <math>d</math> denote the length of <math>\overline{BD},</math> the longer diagonal of <math>ABCD.</math> Then <math>d^2</math> can be written in the form <math>m+n\sqrt p,</math> where <math>m,n,</math> and <math>p</math> are positive integers and <math>p</math> is not divisible by the square of any prime. What is <math>m+n+p?</math><br />
<br />
<math>\textbf{(A) }81 \qquad \textbf{(B) }89 \qquad \textbf{(C) }97\qquad \textbf{(D) }105 \qquad \textbf{(E) }113</math><br />
<br />
[[2021 AMC 12B Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
Let <math>S</math> be the set of lattice points in the coordinate plane, both of whose coordinates are integers between <math>1</math> and <math>30,</math> inclusive. Exactly <math>300</math> points in <math>S</math> lie on or below a line with equation <math>y=mx.</math> The possible values of <math>m</math> lie in an interval of length <math>\frac ab,</math> where <math>a</math> and <math>b</math> are relatively prime positive integers. What is <math>a+b?</math><br />
<br />
<math>\textbf{(A) }31 \qquad \textbf{(B) }47 \qquad \textbf{(C) }62\qquad \textbf{(D) }72 \qquad \textbf{(E) }85</math><br />
<br />
[[2021 AMC 12B Problems/Problem 25|Solution]]<br />
<br />
<br />
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 900 lattice points. I estimate that the slope, m, 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 600 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 are is above the line. \\<br />
<br />
I find that the number of coordinates with <math>x=1</math> above the line is 30, 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 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 <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 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. \\<br />
<br />
To find the upper bound, notice that each point with an integer-valued x-coordinate is either <math>\frac{1}{3}</math> or <math>\frac{2}{3}</math> 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 <math>x=28, 29, 30</math> which the line <math>y=\frac{2}{3}x</math> intersects at <math>y= \frac{56}{3}, \frac{58}{3}, 20</math>. 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 slope of the line which goes through both the origin and (28,19) is <math>y=\frac{19}{28}x</math>. This gives an upper bound of <math>m=\frac{19}{28}</math>. \\<br />
<br />
Taking the upper bound of m and subtracting the lower bound yields <math>\frac{19}{28}-\frac{2}{3}=\frac{1}{84}</math>. This is answer E.<br />
<br />
==See also==<br />
{{AMC12 box|year=2021|ab=B|before=[[2021 AMC 12A Problems]]|after=[[2022 AMC 12A Problems]]}}<br />
<br />
[[Category:AMC 12 Problems]]<br />
{{MAA Notice}}</div>Theajl