Problem D
Grade Curving

An instructor may use curved score to grade a student’s performance. Assume a student’s original score is $x$ out of a total of $100$. One of the curving methods is to apply to $x$ the curving function $f(x) = 10\sqrt {x}$. The function may be applied $k \geq 0$ times. For better score readability, the result is rounded to the next integer and the final curved score is $y = \lceil f^ k(x) \rceil $.

For example, if the curving function is applied $k = 3$ times to $x = 80$, it changes the score to

\begin{align*} f^3(80) & = f^2(f(80)) = f^2(89.4427) \\ & = f(f(89.4427)) = f(94.5742) = 97.2492. \end{align*}

So the curved score is $y = \lceil 97.2492 \rceil = 98$.

John wants to curve a student’s score $x$ using this method so that the curved score $y$ is between $y_{low}$ and $y_{high}$. How many times should he apply the curving function? In particular, what are the minimum and maximum choices of $k$, so that $y_{low} \leq y \leq y_{high}$?


The input has a single line with three integers: $x$, $y_{low}$, and $y_{high}$ ($1 \leq x \leq y_{low} \leq y_{high} \leq 100$).


Output the minimum and maximum possible choices of $k$. If the curving function can be applied an infinite number of times, output “inf” for the maximum. If there is no $k$ that meets the curving requirement, output a single word “impossible”.

Sample Input 1 Sample Output 1
80 85 98
1 3
Sample Input 2 Sample Output 2
98 98 100
0 inf
Sample Input 3 Sample Output 3
80 85 89

Please log in to submit a solution to this problem

Log in