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Difference between revisions of "2020 AMC 8 Problems/Problem 16"

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==Problem==
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==Problem 16==
  
 
Each of the points <math>A,B,C,D,E,</math> and <math>F</math> in the figure below represents a different digit from <math>1</math> to <math>6.</math> Each of the five lines shown passes through some of these points. The digits along each line are added to produce five sums, one for each line. The total of the five sums is <math>47.</math> What is the digit represented by <math>B?</math>
 
Each of the points <math>A,B,C,D,E,</math> and <math>F</math> in the figure below represents a different digit from <math>1</math> to <math>6.</math> Each of the five lines shown passes through some of these points. The digits along each line are added to produce five sums, one for each line. The total of the five sums is <math>47.</math> What is the digit represented by <math>B?</math>
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<math>\textbf{(A) }1 \qquad \textbf{(B) }2 \qquad \textbf{(C) }3 \qquad \textbf{(D) }4 \qquad \textbf{(E) }5</math>
 
<math>\textbf{(A) }1 \qquad \textbf{(B) }2 \qquad \textbf{(C) }3 \qquad \textbf{(D) }4 \qquad \textbf{(E) }5</math>
  
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==Solutions==
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== Solution 1 ==
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===Solution 1===
 
 
We can form the following expressions for the sum along each line:
 
We can form the following expressions for the sum along each line:
 
<cmath>\begin{dcases}A+B+C\\A+E+F\\C+D+E\\B+D\\B+F\end{dcases}</cmath>
 
<cmath>\begin{dcases}A+B+C\\A+E+F\\C+D+E\\B+D\\B+F\end{dcases}</cmath>
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Adding these together, we must have <math>2A+3B+2C+2D+2E+2F=47</math>, i.e. <math>2(A+B+C+D+E+F)+B=47</math>. Since <math>A,B,C,D,E,F</math> are all different integers between <math>1</math> and <math>6</math>, we obtain <math>A+B+C+D+E+F=1+2+3+4+5+6=21</math> (where the order doesn't matter as addition is commutative), so our equation simplifies to <math>42 + B = 47</math>. This means <math>B = \boxed{\textbf{(E) }5}</math>.
+
Adding these together, we must have <math>2A+3B+2C+2D+2E+2F=47</math>, i.e. <math>2(A+B+C+D+E+F)+B=47</math>. Since <math>A,B,C,D,E,F</math> are unique integers between <math>1</math> and <math>6</math>, we obtain <math>A+B+C+D+E+F=1+2+3+4+5+6=21</math> (where the order doesn't matter as addition is commutative), so our equation simplifies to <math>42 + B = 47</math>. This means <math>B = \boxed{\textbf{(E) }5}</math>.
 
~RJ5303707
 
~RJ5303707
  
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===Solution 2===
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== Solution 2 ==
 
Following the first few steps of Solution 1, we have <math>2(A+C+D+E+F)+3B=47</math>. Because an even number (<math>2(A+C+D+E+F)</math>) subtracted from an odd number (47) is always odd, we know that <math>3B</math> is odd, showing that <math>B</math> is odd. Now we know that <math>B</math> is either 1, 3, or 5. If we try <math>B=1</math>, we get <math>43=47</math> which is not true. Testing <math>B=3</math>, we get <math>45=47</math>, which is also not true. Therefore, we have <math>B = \boxed{\textbf{(E) }5}</math>.
 
Following the first few steps of Solution 1, we have <math>2(A+C+D+E+F)+3B=47</math>. Because an even number (<math>2(A+C+D+E+F)</math>) subtracted from an odd number (47) is always odd, we know that <math>3B</math> is odd, showing that <math>B</math> is odd. Now we know that <math>B</math> is either 1, 3, or 5. If we try <math>B=1</math>, we get <math>43=47</math> which is not true. Testing <math>B=3</math>, we get <math>45=47</math>, which is also not true. Therefore, we have <math>B = \boxed{\textbf{(E) }5}</math>.
  
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==Video Solution by North America Math Contest Go Go Go==
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==Video Solution by NiuniuMaths (Easy to understand!)==
 +
https://www.youtube.com/watch?v=bHNrBwwUCMI
 +
 
 +
~NiuniuMaths
 +
 
 +
== Video Solution by Math-X (First understand the problem!!!) ==
 +
https://youtu.be/UnVo6jZ3Wnk?si=n-vOSvxYPmuhVmaq&t=2574
 +
 
 +
~Math-X
 +
 
 +
== Video Solution (CLEVER MANIPULATIONS!!!) ==
 +
https://youtu.be/W8pib6O_6xA
 +
 
 +
~<i>Education, the Study of Everything</i>
 +
 
 +
== Video Solution by North America Math Contest Go Go Go ==
  
 
https://www.youtube.com/watch?v=hwCb64F34XE
 
https://www.youtube.com/watch?v=hwCb64F34XE
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~North America Math Contest Go Go Go
 
~North America Math Contest Go Go Go
  
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==Video Solution by WhyMath==
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== Video Solution by WhyMath ==
 
https://youtu.be/1ldTmo4J7Es
 
https://youtu.be/1ldTmo4J7Es
  
 
~savannahsolver
 
~savannahsolver
  
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==Video Solution==
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== Video Solution ==
 
https://www.youtube.com/watch?v=a3Z7zEc7AXQ
 
https://www.youtube.com/watch?v=a3Z7zEc7AXQ
 
-LOUISGENIUS
 
-LOUISGENIUS
  
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==Video Solution by Interstigation==
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== Video Solution by Interstigation ==
 
https://youtu.be/YnwkBZTv5Fw?t=728
 
https://youtu.be/YnwkBZTv5Fw?t=728
  
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~ pi_is_3.14
 
~ pi_is_3.14
  
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==Video Solution (CLEVER MANIPULATIONS!!!)==
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== See also ==
−
https://youtu.be/W8pib6O_6xA
 
−
 
 
−
~Interstigation
 
−
 
 
−
==See also==
 
 
{{AMC8 box|year=2020|num-b=15|num-a=17}}
 
{{AMC8 box|year=2020|num-b=15|num-a=17}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 20:57, 3 September 2024

Problem 16

Each of the points $A,B,C,D,E,$ and $F$ in the figure below represents a different digit from $1$ to $6.$ Each of the five lines shown passes through some of these points. The digits along each line are added to produce five sums, one for each line. The total of the five sums is $47.$ What is the digit represented by $B?$

[asy] size(200); dotfactor = 10; pair p1 = (-28,0); pair p2 = (-111,213); draw(p1--p2,linewidth(1)); pair p3 = (-160,0); pair p4 = (-244,213); draw(p3--p4,linewidth(1)); pair p5 = (-316,0); pair p6 = (-67,213); draw(p5--p6,linewidth(1)); pair p7 = (0, 68); pair p8 = (-350,10); draw(p7--p8,linewidth(1)); pair p9 = (0, 150); pair p10 = (-350, 62); draw(p9--p10,linewidth(1)); pair A = intersectionpoint(p1--p2, p5--p6); dot("$A$", A, 2*W); pair B = intersectionpoint(p5--p6, p3--p4); dot("$B$", B, 2*WNW); pair C = intersectionpoint(p7--p8, p5--p6); dot("$C$", C, 1.5*NW); pair D = intersectionpoint(p3--p4, p7--p8); dot("$D$", D, 2*NNE); pair EE = intersectionpoint(p1--p2, p7--p8); dot("$E$", EE, 2*NNE); pair F = intersectionpoint(p1--p2, p9--p10); dot("$F$", F, 2*NNE); [/asy]

$\textbf{(A) }1 \qquad \textbf{(B) }2 \qquad \textbf{(C) }3 \qquad \textbf{(D) }4 \qquad \textbf{(E) }5$

Solution 1

We can form the following expressions for the sum along each line: \[\begin{dcases}A+B+C\\A+E+F\\C+D+E\\B+D\\B+F\end{dcases}\] Adding these together, we must have $2A+3B+2C+2D+2E+2F=47$, i.e. $2(A+B+C+D+E+F)+B=47$. Since $A,B,C,D,E,F$ are unique integers between $1$ and $6$, we obtain $A+B+C+D+E+F=1+2+3+4+5+6=21$ (where the order doesn't matter as addition is commutative), so our equation simplifies to $42 + B = 47$. This means $B = \boxed{\textbf{(E) }5}$. ~RJ5303707

Solution 2

Following the first few steps of Solution 1, we have $2(A+C+D+E+F)+3B=47$. Because an even number ($2(A+C+D+E+F)$) subtracted from an odd number (47) is always odd, we know that $3B$ is odd, showing that $B$ is odd. Now we know that $B$ is either 1, 3, or 5. If we try $B=1$, we get $43=47$ which is not true. Testing $B=3$, we get $45=47$, which is also not true. Therefore, we have $B = \boxed{\textbf{(E) }5}$.

Video Solution by NiuniuMaths (Easy to understand!)

https://www.youtube.com/watch?v=bHNrBwwUCMI

~NiuniuMaths

Video Solution by Math-X (First understand the problem!!!)

https://youtu.be/UnVo6jZ3Wnk?si=n-vOSvxYPmuhVmaq&t=2574

~Math-X

Video Solution (CLEVER MANIPULATIONS!!!)

https://youtu.be/W8pib6O_6xA

~Education, the Study of Everything

Video Solution by North America Math Contest Go Go Go

https://www.youtube.com/watch?v=hwCb64F34XE

~North America Math Contest Go Go Go

Video Solution by WhyMath

https://youtu.be/1ldTmo4J7Es

~savannahsolver

Video Solution

https://www.youtube.com/watch?v=a3Z7zEc7AXQ -LOUISGENIUS

Video Solution by Interstigation

https://youtu.be/YnwkBZTv5Fw?t=728

~Interstigation

Video Solution by OmegaLearn

https://youtu.be/sZfOjGtEtEY?t=604

~ pi_is_3.14

See also

2020 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 15 		Followed by
Problem 17
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 AJHSME/AMC 8 Problems and Solutions

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