Nearest Fraction

AtCoder

IDabc333_g

Time2000ms

Memory256MB

Difficulty—

You are given a positive real number $r$ less than $1$, and a positive integer $N$. Among the pair of integers $(p,q)$ such that $0\leq p\leq q\leq N$ and $\gcd(p,q)=1$, find the one that minimizes $\left\vert r-\dfrac pq\right\vert$. If multiple such pairs $(p,q)$ exist, print the one with the smallest value of $\dfrac pq$. ## Constraints * $0\lt r\lt 1$ * $r$ is given as a real number with at most $18$ decimal places. * $1\leq N\leq 10^{10}$ * $N$ is an integer. ## Input The input is given from Standard Input in the following format: $r$ $N$ [samples]

Samples

Input #1

0.333
33

Output #1

1 3

$\left\vert0.333-\dfrac13\right\vert=\dfrac1{3000}$. There is no $0\leq p\leq q\leq33,\gcd(p,q)=1$ such that $\left\vert0.333-\dfrac pq\right\vert\lt\dfrac1{3000}$, so print `1 3`.

Input #2

0.45
5

Output #2

2 5

$\left\vert0.45-\dfrac12\right\vert=\left\vert0.45-\dfrac25\right\vert=\dfrac1{20}$. There is no $0\leq p\leq q\leq5,\gcd(p,q)=1$ such that $\left\vert0.45-\dfrac pq\right\vert\lt\dfrac1{20}$, and we have $\dfrac12\gt\dfrac25$, so print `2 5`.

Input #3

0.314159265358979323
10000

Output #3

71 226

$\left\vert0.314159265358979323-\dfrac{71}{226}\right\vert=\dfrac{3014435336501}{113000000000000000000}$.

Input #4

0.007735339533561113
7203576162

Output #4

34928144 4515398949

API Response (JSON)

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    "name": "Nearest Fraction",
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      "content": "You are given a positive real number $r$ less than $1$, and a positive integer $N$. Among the pair of integers $(p,q)$ such that $0\\leq p\\leq q\\leq N$ and $\\gcd(p,q)=1$, find the one that minimizes $\\",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc333_g"
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  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given a positive real number $r$ less than $1$, and a positive integer $N$.\nAmong the pair of integers $(p,q)$ such that $0\\leq p\\leq q\\leq N$ and $\\gcd(p,q)=1$, find the one that minimizes $\\...",
      "is_translate": false,
      "language": "English"
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