2000 AIME I Problems/Problem 3

Problem

In the expansion of $(ax + b)^{2000},$ where $a$ and $b$ are relatively prime positive integers, the coefficients of $x^{2}$ and $x^{3}$ are equal. Find $a + b$ .

Solution

Using the binomial theorem, $\binom{2000}{2} b^{1998}a = \binom{2000}{3}b^{1997}a^2 \Longrightarrow b=666a$ .

Since $a$ and $b$ are positive relatively prime integers, $a=1$ and $b=666$ , and $a+b=\boxed{667}$ .

2000 AIME I (Problems • Answer Key • Resources)
Preceded by Problem 2	Followed by Problem 4
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
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2000 AIME I Problems/Problem 3

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