# Pyramid with Square base | AIME I, 1995 | Question 12

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1995 based on Pyramid with Square base.

## Pyramid with Squared base - AIME I, 1995

Pyramid OABCD has square base ABCD, congruent edges OA,OB,OC,OD and Angle AOB=45, Let $\theta$ be the measure of dihedral angle formed by faces OAB and OBC, given that cos$\theta$=m+$\sqrt{n}$, find m+n.

• is 107
• is 5
• is 840
• cannot be determined from the given information

### Key Concepts

Integers

Divisibility

Algebra

AIME I, 1995, Question 12

Geometry Vol I to IV by Hall and Stevens

## Try with Hints

First hint

Let $\theta$ be angle formed by two perpendiculars drawn to BO one from plane ABC and one from plane OBC.

Let AP=1 $\Delta$ APO is a right angled isosceles triangle, OP=AP=1.

Second Hint

then OB=OA=$\sqrt{2}$, AB=$\sqrt{4-2\sqrt{2}}$, AC=$\sqrt{8-4\sqrt{2}}$

Final Step

taking cosine law

$AC^{2}=AP^{2}+PC^{2}-2(AP)(PC)cos\theta$

or, 8-4$\sqrt{2}$=1+1-$2cos\theta$ or, cos$\theta$=-3+$\sqrt{8}$

or, m+n=8-3=5.

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