Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2000 based on Algebra and Combination.

## Algebra and combination – AIME 2000

In expansion \((ax+b)^{2000}\) where a and b are relatively prime positive integers the coefficient of \(x^{2}\) and \(x^{3}\) are equal, find a+b

- is 107
- is 667
- is 840
- cannot be determined from the given information

**Key Concepts**

Algebra

Equations

Combination

## Check the Answer

But try the problem first…

Answer: is 667.

AIME, 2000, Question 3

Elementary Algebra by Hall and Knight

## Try with Hints

First hint

here coefficient of \(x^{2}\)= coefficient of \(x^{3}\) in the same expression

Second Hint

then \({2000 \choose 1998}a^{2}b^{1998}\)=\({2000 \choose 1997}a^{3}b^{1997}\)

Final Step

then \(b=\frac{1998}{3}\)a=666a where a and b are relatively prime that is a=1,b=666 then a+b=666+1=667.

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## Other useful links

- https://www.cheenta.com/cubes-and-rectangles-math-olympiad-hanoi-2018/
- https://www.youtube.com/watch?v=ST58GTF95t4&t=140s