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
Check the Answer
Answer: is 5.
AIME I, 1995, Question 12
Geometry Vol I to IV by Hall and Stevens
Try with Hints
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.

then OB=OA=\(\sqrt{2}\), AB=\(\sqrt{4-2\sqrt{2}}\), AC=\(\sqrt{8-4\sqrt{2}}\)
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.
Other useful links
- https://www.cheenta.com/rational-number-and-integer-prmo-2019-question-9/
- https://www.youtube.com/watch?v=lBPFR9xequA