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Difference between revisions of "2024 AMC 10A Problems/Problem 20"

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==Problem==
 
==Problem==
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Let <math>S</math> be a subset of <math>\{1, 2, 3, \dots, 2024\}</math> such that the following two conditions hold:
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Point <math>X</math> lies outside regular pentagon <math>ABCDE</math> so that <math>\triangle BXE</math> is an equilateral triangle, as shown below. What is the degree measure of acute angle <math>\angle CXD</math>?
  
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*If <math>x</math> and <math>y</math> are distinct elements of <math>S</math>, then <math>|x-y| > 2.</math>
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<asy>
 +
import graph; size(7cm);
 +
real labelscalefactor = 0.75; /* changes label-to-point distance */
 +
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
 +
pen dotstyle = black; /* point style */
 +
real xmin = -1.089556028373145, xmax = 2.738320502950249, ymin = -1.3143096399343266, ymax = 1.2691431521185288; /* image dimensions */
 +
pen qqwuqq = rgb(0,0.39215686274509803,0);
  
−
*If <math>x</math> and <math>y</math> are distinct odd elements of <math>S</math>, then <math>|x-y| > 6.</math>
+
filldraw(arc((1.9562952014676112,0),0.25,168,192)--(1.9562952014676112,0)--cycle, mediumgrey);
 +
/* draw figures */
 +
draw((-0.8090169943749473,0.5877852522924731)--(0.30901699437494745,0.9510565162951535));
 +
draw((0.30901699437494745,0.9510565162951535)--(1,0));
 +
draw((1,0)--(0.3090169943749473,-0.9510565162951536));
 +
draw((0.3090169943749473,-0.9510565162951536)--(-0.8090169943749475,-0.587785252292473));
 +
draw((-0.8090169943749475,-0.587785252292473)--(-0.8090169943749473,0.5877852522924731));
 +
draw((0.30901699437494745,0.9510565162951535)--(0.3090169943749473,-0.9510565162951536));
 +
draw((0.3090169943749473,-0.9510565162951536)--(1.9562952014676112,0));
 +
draw((1.9562952014676112,0)--(0.30901699437494745,0.9510565162951535));
 +
draw((-0.8090169943749473,0.5877852522924731)--(1.9562952014676112,0), dashed);
 +
draw((1.9562952014676112,0)--(-0.8090169943749475,-0.587785252292473), dashed);
 +
/* dots and labels */
 +
dot((1,0),linewidth(4pt) + dotstyle);
 +
label("$A$", (1.013592864312142,0), E * labelscalefactor);
 +
dot((0.30901699437494745,0.9510565162951535),linewidth(4pt) + dotstyle);
 +
label("$B$", (0.26,1), NE * labelscalefactor);
 +
dot((-0.8090169943749473,0.5877852522924731),linewidth(4pt) + dotstyle);
 +
label("$C$", (-0.82,0.64), NW * 0.25);
 +
dot((-0.8090169943749475,-0.587785252292473),linewidth(4pt) + dotstyle);
 +
label("$D$", (-0.82,-0.64), SW * 0.25);
 +
dot((0.3090169943749473,-0.9510565162951536),linewidth(4pt) + dotstyle);
 +
label("$E$", (0.26,-1), SE * labelscalefactor);
 +
dot((1.9562952014676112,0),linewidth(4pt) + dotstyle);
 +
label("$X$", (1.9705619971429902, 0), E * labelscalefactor);
 +
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
 +
/* end of picture */
 +
</asy>
  
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What is the maximum possible number of elements in <math>S</math>?
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<math>\textbf{(A)}~18^{\circ}\qquad\textbf{(B)}~19.5^{\circ}\qquad\textbf{(C)}~21^{\circ}\qquad\textbf{(D)}~22.5^{\circ}\qquad\textbf{(E)}~24^{\circ}</math>
  
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<math>\textbf{(A) }436 \qquad \textbf{(B) }506 \qquad \textbf{(C) }608 \qquad \textbf{(D) }654 \qquad \textbf{(E) }675</math>
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==Solution==
 +
Note that <math>EB = EC = EX</math>, so it follows that <math>E</math> is the circumcenter of <math>\triangle BCX</math>. Therefore, <math>\angle BXC = \tfrac{\angle BEC}{2} = 18^{\circ}</math>, so by symmetry <cmath>\angle CXD = \angle BXE - 2 \angle BXC = 60^{\circ} - 2 \cdot 18^{\circ} = \boxed{\textbf{(E)}~24^{\circ}}.</cmath>
  
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==Video Solution by Scholars Foundation==
+
~ihatemath123
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https://www.youtube.com/watch?v=FKOqZau--5w&t=1s
 
−
 
 
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==Solution 1==
 
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All lists are organized in ascending order:
 
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By listing out the smallest possible elements of subset <math>S,</math> we can find that subset <math>S</math> starts with <math>\{1, 4, 8, 11, 14, 18, 21, 24, 28, 31, \dots\}.</math> It is easily noticed that the elements of the subset "loop around" every 3 elements, specifically adding 10 each time. This means that there will be <math>2024/10</math> or <math>202R4</math> whole loops in the subset <math>S,</math> implying that there will be <math>202*3 = 606</math> elements in S. However, we have undercounted, as we did not count the remainder that resulted from <math>2024/10</math><math>.</math> With a remainder of <math>4,</math> we can fit <math>2</math> more elements into the subset <math>S,</math> namely <math>2021</math> and <math>2024,</math> resulting in a total of <math>606+2</math> or <math>\boxed{\textbf{(C) }608}</math> elements in subset <math>S</math>.
 
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NOTE:
 
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To prove that this is the best we can do, consider adding each element one by one, for the first element, say n. If n is greater than 2, we can choose n - 2 which is always better. Therefore, n = 1 or n = 2.
 
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If n = 2 was optimal, then choose it, then the set of usable numbers in <math>S</math> becomes 5 through 2024. We can transform the usable set of <math>S</math> to <math>Q</math> where <math>Q</math> contains the numbers 1 through 2020. Because we assumed n = 2 was optimal, we can choose n = 2 for the set <math>Q</math> too. Because every element in <math>Q</math> is 4 below the elements of <math>S</math>, choosing 2 in <math>Q</math> would mean choosing 6 in set <math>S</math>. By induction we see that our list would be {2,6,10,14,18,.....2022} which only gives 506 elements which is sub-optimal. Therefore, we can conclude that n = 1 is optimal, and we proceed as the solution above.
 
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-weihou0
 
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==Solution 2==
 
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Notice that we can first place odd numbers, then place even numbers between each pair. We can start at <math>1</math> and continue from there. Realize that the smallest number <math>k</math> such that <math>kx+1</math> reproduces odd number is <math>8</math>. The next ones are <math>10, 12, 14</math>. We can proceed to find the number of numbers in this particular sequence. From the equation <math>8x+1=2023</math>, we get that <math>x \leq 252.875</math> works, so this means there is 253 solutions. Looking at <math>1,2,3,4,5,6,7,9</math> we can see that there could only be 1 possible number between each pair, yielding <math>252+253=505</math>. Then see that we can fit two more into the number count since the set <math>2017</math> to <math>2024</math> can fit two evens. Now this means <math>A</math> and <math>B</math> don’t work. Now test out <math>10x+1</math>. Using the same method, we get that <math>608</math> is the maximum number in the set. Everything above <math>x=10</math> doesn’t work, as we can split it down into smaller subgroups, so the answer is <math>\boxed{\textbf{(C) }608}</math>.
 
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−
~EaZ_Shadow
 
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== Video Solution by Power Solve ==
 
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https://www.youtube.com/watch?v=NZ0SBMqeAfg
 
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== Video Solution by Pi Academy ==
 
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https://youtu.be/fW7OGWee31c?si=oq7toGPh2QaksLHE
 
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==Video Solution by SpreadTheMathLove==
 
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https://www.youtube.com/watch?v=6SQ74nt3ynw
 
  
 
==See also==
 
==See also==
 
{{AMC10 box|year=2024|ab=A|num-b=19|num-a=21}}
 
{{AMC10 box|year=2024|ab=A|num-b=19|num-a=21}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 21:24, 20 March 2025

Problem

Point $X$ lies outside regular pentagon $ABCDE$ so that $\triangle BXE$ is an equilateral triangle, as shown below. What is the degree measure of acute angle $\angle CXD$?

[asy] import graph; size(7cm); real labelscalefactor = 0.75; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -1.089556028373145, xmax = 2.738320502950249, ymin = -1.3143096399343266, ymax = 1.2691431521185288; /* image dimensions */ pen qqwuqq = rgb(0,0.39215686274509803,0); filldraw(arc((1.9562952014676112,0),0.25,168,192)--(1.9562952014676112,0)--cycle, mediumgrey); /* draw figures */ draw((-0.8090169943749473,0.5877852522924731)--(0.30901699437494745,0.9510565162951535)); draw((0.30901699437494745,0.9510565162951535)--(1,0)); draw((1,0)--(0.3090169943749473,-0.9510565162951536)); draw((0.3090169943749473,-0.9510565162951536)--(-0.8090169943749475,-0.587785252292473)); draw((-0.8090169943749475,-0.587785252292473)--(-0.8090169943749473,0.5877852522924731)); draw((0.30901699437494745,0.9510565162951535)--(0.3090169943749473,-0.9510565162951536)); draw((0.3090169943749473,-0.9510565162951536)--(1.9562952014676112,0)); draw((1.9562952014676112,0)--(0.30901699437494745,0.9510565162951535)); draw((-0.8090169943749473,0.5877852522924731)--(1.9562952014676112,0), dashed); draw((1.9562952014676112,0)--(-0.8090169943749475,-0.587785252292473), dashed); /* dots and labels */ dot((1,0),linewidth(4pt) + dotstyle); label("$A$", (1.013592864312142,0), E * labelscalefactor); dot((0.30901699437494745,0.9510565162951535),linewidth(4pt) + dotstyle); label("$B$", (0.26,1), NE * labelscalefactor); dot((-0.8090169943749473,0.5877852522924731),linewidth(4pt) + dotstyle); label("$C$", (-0.82,0.64), NW * 0.25); dot((-0.8090169943749475,-0.587785252292473),linewidth(4pt) + dotstyle); label("$D$", (-0.82,-0.64), SW * 0.25); dot((0.3090169943749473,-0.9510565162951536),linewidth(4pt) + dotstyle); label("$E$", (0.26,-1), SE * labelscalefactor); dot((1.9562952014676112,0),linewidth(4pt) + dotstyle); label("$X$", (1.9705619971429902, 0), E * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy]

$\textbf{(A)}~18^{\circ}\qquad\textbf{(B)}~19.5^{\circ}\qquad\textbf{(C)}~21^{\circ}\qquad\textbf{(D)}~22.5^{\circ}\qquad\textbf{(E)}~24^{\circ}$

Solution

Note that $EB = EC = EX$, so it follows that $E$ is the circumcenter of $\triangle BCX$. Therefore, $\angle BXC = \tfrac{\angle BEC}{2} = 18^{\circ}$, so by symmetry \[\angle CXD = \angle BXE - 2 \angle BXC = 60^{\circ} - 2 \cdot 18^{\circ} = \boxed{\textbf{(E)}~24^{\circ}}.\]

~ihatemath123

See also

2024 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 19 		Followed by
Problem 21
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. AMC Logo.png

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