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AIME I Geometry Math Olympiad USA Math Olympiad

Cross section of solids and volumes | AIME I 2012 | Question 8

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2012 based on Cross section of solids and volumes.

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2012 based on cross section of solids and volumes.

Cross-section of solids and volumes – AIME I, 2012


Cube ABCDEFGH labeled as shown below has edge length 1 and is cut by a plane passing through vertex D and the midpoints M and N of AB and CG respectively. The plane divides the cube into solids. The volume of the larger of the two solids can be written in the form \(\frac{p}{q}\) where p and q are relatively prime. find p+q.

Cross section of solids and volumes
  • is 107
  • is 89
  • is 840
  • cannot be determined from the given information

Key Concepts


Calculus

Algebra

Geometry

Check the Answer


But try the problem first…

Answer: 89.

Source
Suggested Reading

AIME, 2012, Question 8

Calculus Vol 1 and 2 by Apostle

Try with Hints


First hint

DMN plane cuts the section of solid with \(z=\frac{y}{2}-\frac{x}{4}\) intersects base at \(y=\frac{x}{2}\)

Second Hint

\(V=\int_0^1\int_{\frac{x}{2}}^1\int_0^{\frac{y}{2}-\frac{x}{4}}{d}x{d}y{d}z\)=\(\frac{7}{48}\)

Final Step

other portion 1-\(\frac{7}{48}\)=\(\frac{41}{48}\) then 41+48=89.

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