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Algebra Difficulty 4.9 AIME Prove it United States

Problem:

Let $f$ be a function from the nonnegative integers to the positive reals such that $f(x+y) = f(x) \cdot f(y)$ holds for all nonnegative integers $x$ and $y$ . If $f(19) = 524288 k$ , find $f(4)$ in terms of $k$ .

Solution

Solution:

Answer: $16 k^{4 / 19}$

The given condition implies $f(m n) = f(m)^n$ , so
$f(4)^{19} = f(4 \cdot 19) = f(19 \cdot 4) = f(19)^4$
and it follows that $f(4) = 16 k^{4 / 19}$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.