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26

Part of 1994 AMC 12/AHSME

Problems(1)

1994 AMC 12 #26

Source:

12/30/2011
A regular polygon of mm sides is exactly enclosed (no overlaps, no gaps) by mm regular polygons of nn sides each. (Shown here for m=4,n=8m=4, n=8.) If m=10m=10, what is the value of nn? [asy] size(200); defaultpen(linewidth(0.8)); draw(unitsquare); path p=(0,1)--(1,1)--(1+sqrt(2)/2,1+sqrt(2)/2)--(1+sqrt(2)/2,2+sqrt(2)/2)--(1,2+sqrt(2))--(0,2+sqrt(2))--(-sqrt(2)/2,2+sqrt(2)/2)--(-sqrt(2)/2,1+sqrt(2)/2)--cycle; draw(p); draw(shift((1+sqrt(2)/2,-sqrt(2)/2-1))*p); draw(shift((0,-2-sqrt(2)))*p); draw(shift((-1-sqrt(2)/2,-sqrt(2)/2-1))*p);[/asy] <spanclass=′latex−bold′>(A)</span> 5<spanclass=′latex−bold′>(B)</span> 6<spanclass=′latex−bold′>(C)</span> 14<spanclass=′latex−bold′>(D)</span> 20<spanclass=′latex−bold′>(E)</span> 26 <span class='latex-bold'>(A)</span>\ 5 \qquad<span class='latex-bold'>(B)</span>\ 6 \qquad<span class='latex-bold'>(C)</span>\ 14 \qquad<span class='latex-bold'>(D)</span>\ 20 \qquad<span class='latex-bold'>(E)</span>\ 26
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