Difference between revisions of "2020 AMC 8 Problems/Problem 25"
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As the diagram shows, <math>s_{1}+s_{2}+s_{3}=3322, s_{1}-s_{2}+s_{3}=2020.</math> If we subtract the second equation from the first we will get <math>2s_{2}=1302</math> or <math>s_{2}=\boxed{651\textbf{(A)}}</math> ~icematrix, edits by starrynight7210 | As the diagram shows, <math>s_{1}+s_{2}+s_{3}=3322, s_{1}-s_{2}+s_{3}=2020.</math> If we subtract the second equation from the first we will get <math>2s_{2}=1302</math> or <math>s_{2}=\boxed{651\textbf{(A)}}</math> ~icematrix, edits by starrynight7210 | ||
| + | |||
| + | ==Solution 2== | ||
| + | |||
| + | WLOG, assume that <math>S_1=S_3</math> and <math>R_1=R_2</math>. Let the sum of the lengths of <math>S_1</math> and <math>S_2</math> be <math>x</math> and let the length of <math>S_2</math> be <math>y</math>. We have the system \begin{align*} | ||
| + | x+y&=3322\\ | ||
| + | x-y&=2020 | ||
| + | \end{align*} | ||
| + | which we solve to find that <math>y=\textbf{(A) }651</math>. | ||
| + | |||
| + | -franzliszt | ||
==See also== | ==See also== | ||
{{AMC8 box|year=2020|num-b=24|after=Last Problem}} | {{AMC8 box|year=2020|num-b=24|after=Last Problem}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 14:11, 18 November 2020
Rectangles 
and 
and squares 
and 
shown below, combine to form a rectangle that is 3322 units wide and 2020 units high. What is the side length of 
in units?
![[asy] draw((0,0)--(5,0)--(5,3)--(0,3)--(0,0)); draw((3,0)--(3,1)--(0,1)); draw((3,1)--(3,2)--(5,2)); draw((3,2)--(2,2)--(2,1)--(2,3)); label("$R_1$",(3/2,1/2)); label("$S_3$",(4,1)); label("$S_2$",(5/2,3/2)); label("$S_1$",(1,2)); label("$R_2$",(7/2,5/2)); [/asy]](http://latex.artofproblemsolving.com/a/5/5/a55c518a1179d67520c6a03827b4ba25ca1413f2.png)
Solution 1
For each square 
, let the sidelength of this square be denoted by 
.
As the diagram shows, 
If we subtract the second equation from the first we will get 
or 
~icematrix, edits by starrynight7210
Solution 2
WLOG, assume that 
and 
. Let the sum of the lengths of 
and be 
and let the length of be 
. We have the system \begin{align*}
x+y&=3322\\ x-y&=2020
\end{align*}
which we solve to find that 
.
-franzliszt
See also
| 2020 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 24 |
Followed by Last Problem | |
| 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. 
