| {"messages": [{"role": "user", "content": "Let $f(n)$ and $g(n)$ be functions satisfying \\[f(n) = \\begin{cases} \\sqrt{n} & \\text{ if } \\sqrt{n} \\text{ is an integer}\\\\ 1 + f(n+1) & \\text{ otherwise} \\end{cases}\\] and \\[g(n) = \\begin{cases}\\sqrt{n} & \\text{ if } \\sqrt{n} \\text{ is an integer}\\\\ 2 + g(n+2) & \\text{ otherwise} \\end{cases}\\] for positive integers $n$ . Find the least positive integer $n$ such that $\\tfrac{f(n)}{g(n)} = \\tfrac{4}{7}$ ."}, {"role": "assistant", "content": "258"}], "solution": "258"} | |