# 2007 AMC 10B Problems/Problem 5

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## Problem

In a certain land, all Arogs are Brafs, all Crups are Brafs, all Dramps are Arogs, and all Crups are Dramps. Which of the following statements is implied by these facts? $\textbf{(A) } \text{All Dramps are Brafs and are Crups.}\\ \textbf{(B) } \text{All Brafs are Crups and are Dramps.}\\ \textbf{(C) } \text{All Arogs are Crups and are Dramps.}\\ \textbf{(D) } \text{All Crups are Arogs and are Brafs.}\\ \textbf{(E) } \text{All Arogs are Dramps and some Arogs may not be Crups.}$

## Solution $[asy] unitsize(6mm); defaultpen(linewidth(.8pt)+fontsize(9pt)); dotfactor=4; real r1=1, r2=2, r3=3, r4=4; pair O1=(0,0), O2=(0,-0.5), O3=(0,-1), O4=(0,-1.5); path circleA=Circle(O1,r1); draw(circleA); path circleB=Circle(O2,r2); draw(circleB); path circleC=Circle(O3,r3); draw(circleC); path circleD=Circle(O4,r4); draw(circleD); label("Crups",(0,-.5)); label("Dramps",(0,-2)); label("Arogs",(0,-3.5)); label("Brafs",(0,-5)); [/asy]$

It may be easier to visualize this by drawing some sort of diagram. From the first statement, you can draw an Arog circle inside of the Braf circle, since all Arogs are Brafs, but not all Brafs are Arogs. Ignore the second statement for now, and draw a Dramp circle in the Arog circle and a Crup circle in the Dramp circle. You can see the second statement is already true because all Crups are Arogs. As you can see, the only statement that is true is $\boxed{\mathrm{(D)}}$

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