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.

- 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.

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.

## Other useful links

- https://www.cheenta.com/cubes-and-rectangles-math-olympiad-hanoi-2018/
- https://www.youtube.com/watch?v=ST58GTF95t4&t=140s

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