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Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1994 based on Points of Equilateral triangle.

The points (0,0), (a,11), and (b,37) are the vertices of equilateral triangle, find the value of ab.

- is 107
- is 315
- is 840
- cannot be determined from the given information

Integers

Complex Number

Equilateral Triangle

But try the problem first...

Answer: is 315.

Source

Suggested Reading

AIME I, 1994, Question 8

Complex Numbers from A to Z by Titu Andreescue

First hint

Let points be on complex plane as b+37i, a+11i and origin.

Second Hint

then \((a+11i)cis60=(a+11i)(\frac{1}{2}+\frac{\sqrt{3}i}{2})\)=b+37i

Final Step

equating real parts b=\(\frac{a}{2}-\frac{11\sqrt{3}}{2}\) is first equation

equating imaginary parts 37=\(\frac{11}{2}+\frac{a\sqrt{3}i}{2}\) is second equation

solving both equations a=\(21\sqrt{3}\), b=\(5\sqrt{3}\)

ab=315.

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

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