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Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1999 based on Squares and triangles.

The two squares share the same centre O and have sides of length 1, The length of AB is \(\frac{43}{99}\) and the area of octagon ABCDEFGH is \(\frac{m}{n}\) where m and n are relatively prime positive integers, find m+n.

- is 107
- is 185
- is 840
- cannot be determined from the given information

Squares

Triangles

Algebra

But try the problem first...

Answer: is 185.

Source

Suggested Reading

AIME I, 1999, Question 4

Geometry Vol I to IV by Hall and Stevens

First hint

Triangle AOB, triangleBOC, triangleCOD, triangleDOE, triangleEOF, triangleFOG, triangleGOH, triangleHOA are congruent triangles

Second Hint

with each area =\(\frac{\frac{43}{99} \times \frac{1}{2}}{2}\)

Final Step

then the area of all 8 of them is (8)\(\frac{\frac{43}{99} \times \frac{1}{2}}{2}\)=\(\frac{86}{99}\) then 86+99=185.

- https://www.cheenta.com/rational-number-and-integer-prmo-2019-question-9/
- https://www.youtube.com/watch?v=lBPFR9xequA

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