Difference between revisions of "2012 AIME II Problems/Problem 5"
Williamhu888 (talk | contribs) (Created page with "== Problem 5 == In the accompanying figure, the outer square <math>S</math> has side length <math>40</math>. A second square <math>S'</math> of side length <math>15</math> is con...") |
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draw(Sp1--M1--Sp2--M2--Sp3--M3--Sp4--M4--cycle); | draw(Sp1--M1--Sp2--M2--Sp3--M3--Sp4--M4--cycle); | ||
</asy></center> | </asy></center> | ||
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| + | == Solution == | ||
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| + | == See also == | ||
| + | {{AIME box|year=2012|n=II|num-b=4|num-a=6}} | ||
Revision as of 16:20, 31 March 2012
Problem 5
In the accompanying figure, the outer square 
has side length 
. A second square 
of side length 
is constructed inside with the same center as and with sides parallel to those of . From each midpoint of a side of , segments are drawn to the two closest vertices of . The result is a four-pointed starlike figure inscribed in . The star figure is cut out and then folded to form a pyramid with base . Find the volume of this pyramid.
![[asy] pair S1 = (20, 20), S2 = (-20, 20), S3 = (-20, -20), S4 = (20, -20); pair M1 = (S1+S2)/2, M2 = (S2+S3)/2, M3=(S3+S4)/2, M4=(S4+S1)/2; pair Sp1 = (7.5, 7.5), Sp2=(-7.5, 7.5), Sp3 = (-7.5, -7.5), Sp4 = (7.5, -7.5); draw(S1--S2--S3--S4--cycle); draw(Sp1--Sp2--Sp3--Sp4--cycle); draw(Sp1--M1--Sp2--M2--Sp3--M3--Sp4--M4--cycle); [/asy]](http://latex.artofproblemsolving.com/2/8/6/2862b9fac9f2c88c10b30e3908cf4ac1d5f62115.png)
Solution
See also
| 2012 AIME II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 4 |
Followed by Problem 6 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
