Problem G
Rationalization
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Having recently read about Pythagoras, Doris has become convinced that there are no irrational numbers! She claims that every number can be written as the quotient of two whole numbers $A$ and $B$. As an exercise, she has decided to find these ratios for some positive physical constants. Because every constant has some error $F$, she only wants a ratio with a quotient between $C-F$ and $C+F$. Furthermore, since she dislikes long numbers, she wants to find the pair $A$, $B$, whose quotient lies within the interval and has the smallest $A$. If there are many such $A$, she chooses the pair $A$, $B$ with smaller $B$. Can you help her accomplish this goal?
Input
The first contains two positive real numbers $C, F$, ($0 < C \leq 10$), ($10^{-8} \leq F \leq 0.1$), the desired quotient and the margin for error, both in decimal form with at most 6 significant digits.
Output
Print two lines with one positive integer each, $A$ and $B$. It is guaranteed that these are less than $10^6$.
Grading
Your solution will be tested on a number of test-case groups. To receive points for a group, your solution must correctly solve every test-case in the group.
|
Group |
Point value |
Restrictions |
|
$1$ |
$20$ |
$B = 1000$ |
|
$2$ |
$40$ |
$A, B \leq 10^{3}$ |
|
$3$ |
$40$ |
No further restrictions |
| Sample Input 1 | Sample Output 1 |
|---|---|
0.01 0.001 |
1 91 |
| Sample Input 2 | Sample Output 2 |
|---|---|
4.71344 0.0000091295 |
1579 335 |
| Sample Input 3 | Sample Output 3 |
|---|---|
0.333 0.00000001 |
333 1000 |
