# 2020 AMC 8 Problems/Problem 25

Rectangles $R_1$ and $R_2,$ and squares $S_1,\,S_2,\,$ and $S_3,$ shown below, combine to form a rectangle that is 3322 units wide and 2020 units high. What is the side length of $S_2$ in units?

$[asy] draw((0,0)--(5,0)--(5,3)--(0,3)--(0,0)); draw((3,0)--(3,1)--(0,1)); draw((3,1)--(3,2)--(5,2)); draw((3,2)--(2,2)--(2,1)--(2,3)); label("R_1",(3/2,1/2)); label("S_3",(4,1)); label("S_2",(5/2,3/2)); label("S_1",(1,2)); label("R_2",(7/2,5/2)); [/asy]$

$\textbf{(A) }651 \qquad \textbf{(B) }655 \qquad \textbf{(C) }656 \qquad \textbf{(D) }662 \qquad \textbf{(E) }666$

## Solution 1

For each square $S_{k}$, let the sidelength of this square be denoted by $s_{k}$.

As the diagram shows, $s_{1}+s_{2}+s_{3}=3322, s_{1}-s_{2}+s_{3}=2020.$ We subtract the second equation from the first, getting $2s_{2}=1302$, and thus $s_{2}=651$, so the answer is $\boxed{\textbf{(A)}\text{ }651}$ ~icematrix, edits by starrynight7210

## Solution 2

WLOG, assume that $S_1=S_3$ and $R_1=R_2$. Let the sum of the lengths of $S_1$ and $S_2$ be $x$ and let the length of $S_2$ be $y$. We have the system $$x+y =3322$$ $$x-y=2020$$

which we solve to find that $y=\textbf{(A) }651$.

-franzliszt

## Solution 3

Since each pair of boxes has a sum of $3322$ or $2020$ and a difference of $S_2$, we see that the answer is $\dfrac{3322 - 2020}{2} = \dfrac{1302}{2} = \boxed{(\text{A}) 651}.$

-A_MatheMagician. Note this is just a more quicker way to do it to get \boxed{(\text{A}) 651}.\$ https://artofproblemsolving.com/community/my-aops

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