Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1995 based on Pyramid with Square base.
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.
Integers
Divisibility
Algebra
Answer: is 5.
AIME I, 1995, Question 12
Geometry Vol I to IV by Hall and Stevens
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.
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