What is the NO-SHORTCUT approach for learning great Mathematics?

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Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1988 based on Complex Plane.

Let w_1,w_2,....,w_n be complex numbers. A line L in the complex plane is called a mean line for the points w_1,w_2,....w_n if L contains points (complex numbers) z_1,z_2, .....z_n such that \(\sum_{k=1}^{n}(z_{k}-w_{k})=0\) for the numbers \(w_1=32+170i, w_2=-7+64i, w_3=-9+200i, w_4=1+27i\) and \(w_5=-14+43i\), there is a unique mean line with y-intercept 3. Find the slope of this mean line.

- is 107
- is 163
- is 634
- cannot be determined from the given information

Integers

Equations

Algebra

But try the problem first...

Answer: is 163.

Source

Suggested Reading

AIME I, 1988, Question 11

Elementary Algebra by Hall and Knight

First hint

\(\sum_{k=1}^{5}w_k=3+504i\)

and \(\sum_{k-1}^{5}z_k=3+504i\)

Second Hint

taking the numbers in the form a+bi

\(\sum_{k=1}^{5}a_k=3\) and \(\sum_{k=1}^{5}b_k=504\)

Final Step

or, y=mx+3 where \(b_k=ma_k+3\) adding all 5 equations given for each k

or, 504=3m+15

or, m=163.

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

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