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Problem 602

AIME late

Algebra Difficulty 5.0 Find the answer

5. Find all solutions to the equation $2017^{x}-2016^{x}=1$ .

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Your answer

Official solution

5. It is obvious that $x=1$ will be a solution to the given equation. Let's show that there are no others. We have: $\left(\frac{2017}{2016}\right)^{x}=1+\frac{1}{2016^{x}} ;\left(\frac{2017}{2016}\right)^{x}-1=\frac{1}{2016^{x}}$ From this, it is clear that the function on the left side is increasing, while the function on the right side is decreasing. Therefore, the equation has no more than one solution. Answer: $\{1\}$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.