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AIME I Algebra Arithmetic Geometry Math Olympiad USA Math Olympiad

Squares and Triangles | AIME I, 1999 | Question 4

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

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

Squares and triangles – AIME I, 1999


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.

Squares and Triangles
  • is 107
  • is 185
  • is 840
  • cannot be determined from the given information

Key Concepts


Squares

Triangles

Algebra

Check the Answer


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

Try with Hints


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.

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