B. Racetrack

Codeforces

IDCF10221B

Time1000ms

Memory64MB

Difficulty—

English · Original

Formal · Original

Alice and Bob play different games. When Alice plays, she always wins exactly a points. When Bob plays, he always wins exactly b points. Today, after they finished playing, they noticed they had the same number of points. What is the smallest number this could be? The first line contains two integers, a and b, separated by spaces, where a is the number of points Alice wins in one game and b is the number of points Bob wins in one game. You should return the smallest possible number of points that Alice and Bob have, which should be an integer c. #cf_span(class=[tex-font-style-underline], body=[Constraints]): 1 ≤ a ≤ 10, 000 1 ≤ b ≤ 10, 000 1 ≤ c ≤ 100, 000, 000 ## Input The first line contains two integers, a and b, separated by spaces, where a is the number of points Alice wins in one game and b is the number of points Bob wins in one game. ## Output You should return the smallest possible number of points that Alice and Bob have, which should be an integer c. [samples] ## Note #cf_span(class=[tex-font-style-underline], body=[Constraints]):1 ≤ a ≤ 10, 0001 ≤ b ≤ 10, 0001 ≤ c ≤ 100, 000, 000

**Definitions** Let $ M \in \mathbb{Z}^+ $ be the number of piles, $ K \in \mathbb{Z}^+ $ the number of distinct pile sizes. Let $ C = \{c_1, c_2, \dots, c_K\} \subset \mathbb{Z}^+ $ be the set of possible pile sizes. **Constraints** 1. $ 1 \le M \le 10^9 $ 2. $ 1 \le K < 2^{17} $ 3. $ 1 \le c_i < 2^{17} $ for all $ i \in \{1, \dots, K\} $ 4. Each pile size is chosen independently and uniformly from $ C $. **Objective** Compute the probability that the Nim position with $ M $ piles, each independently sampled uniformly from $ C $, is a winning position for the first player (Alice). In Nim, a position is winning iff the XOR-sum (nim-sum) of all pile sizes is nonzero. Let $ \mathcal{P} $ be the probability that the nim-sum of $ M $ independent uniform samples from $ C $ is nonzero. Output $ \mathcal{P} \mod 998244353 $, represented as $ P \cdot Q^{-1} \mod 998244353 $, where $ \mathcal{P} = \frac{P}{Q} $ in lowest terms.

API Response (JSON)

{
  "problem": {
    "name": "B. Racetrack",
    "description": {
      "content": "Alice and Bob play different games. When Alice plays, she always wins exactly a points. When Bob plays, he always wins exactly b points.  Today, after they finished playing, they noticed they had the",
      "description_type": "Markdown"
    },
    "platform": "Codeforces",
    "limit": {
      "time_limit": 1000,
      "memory_limit": 65536
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "CF10221B"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Alice and Bob play different games. When Alice plays, she always wins exactly a points. When Bob plays, he always wins exactly b points. \n\nToday, after they finished playing, they noticed they had the...",
      "is_translate": false,
      "language": "English"
    },
    {
      "statement_type": "Markdown",
      "content": "**Definitions**  \nLet $ M \\in \\mathbb{Z}^+ $ be the number of piles, $ K \\in \\mathbb{Z}^+ $ the number of distinct pile sizes.  \nLet $ C = \\{c_1, c_2, \\dots, c_K\\} \\subset \\mathbb{Z}^+ $ be the set of...",
      "is_translate": false,
      "language": "Formal"
    }
  ]
}

Full JSON Raw Segments