2020 AMC 8 Problems/Problem 16
Each of the points 
and 
in the figure below represents a different digit from 
to 
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 
What is the digit represented by 
![[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]](http://latex.artofproblemsolving.com/d/8/7/d878779f0718f46b33d37199ff7b7df96df2948e.png)
Solution 1
We can form the following expressions based on the points in the figure and from the information we are given.
![\[A+B+C\]](http://latex.artofproblemsolving.com/4/7/0/470edd693585a57336d6ad68e697a36b4c52f4d3.png)
![\[A+E+F\]](http://latex.artofproblemsolving.com/8/1/e/81e6aa10b99dc5bca407f1183ef6172d02a4d4ea.png)
![\[C+D+E\]](http://latex.artofproblemsolving.com/6/5/5/65517e53bd9559867e23dae5db4d0ebab1e28648.png)
![\[B+D\]](http://latex.artofproblemsolving.com/d/a/f/daf88f2e594f1653c0f969555871055e827f5ae1.png)
![\[B+F\]](http://latex.artofproblemsolving.com/3/2/7/327e852e65e45622d92b2917adb65a20075fa35d.png)
When we add the five expressions together, and equate it to 47, we get
![\[2A+3B+2C+2D+2E+2F=47.\]](http://latex.artofproblemsolving.com/8/d/3/8d334af43ec0a103c20c5a2b553dcb47b689744d.png)
![\[2(A+B+C+D+E+F)+B=47.\]](http://latex.artofproblemsolving.com/0/2/7/027f87beb10d4de39722bc310049838a97783d13.png)
In addition, we are given that 
, where we can assign the values for A-F randomly because we don't know their individual values. Substituting in our equation, we have
![\[2(21)+B=47.\]](http://latex.artofproblemsolving.com/e/e/6/ee6b561b10ac0f0066df862b07ec9703b7be0c7f.png)
![\[42+B=47\]](http://latex.artofproblemsolving.com/0/7/2/0728416484031d0bffdf91e7927072d804f43443.png)
![\[\boxed{\textbf{(E) }5}\]](http://latex.artofproblemsolving.com/f/4/3/f4341a45c87bb465030a828501595762ef77e21a.png)
~samrocksnature and RJ5303707
Solution 2
We can write an equation. We are given 
which simplifies to 
. Recall that each of 
are unique integers from to 
. Hence, our equation simplifies to 
regardless of which letters equal which numbers. Now we can easily see that the answer is 
.
-franzlizt
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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
