INTRODUCING 5 - days-a-week problem solving session for Math Olympiad and ISI Entrance. Learn More 

April 2, 2020

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.

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




Check the Answer

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


Final Step

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

Subscribe to Cheenta at Youtube

Leave a Reply

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.