F. Drawing cards

Codeforces

IDCF10230F

Time1000ms

Memory256MB

Difficulty—

English · Original

Formal · Original

You have a box with N cards numbered 1...N and you can also create new cards. You are going to draw cards from the box, every time you draw a card of a number that has already been drawn you put that card back into the box. If you draw a card of a number that has not been drawn before, you lay it on the desk and you create another card to put back into the box. The number of the new card is chosen randomly with the same probability from the interval [1, N]. In this problem you have to answer what is the expected number of cards on the desk when you draw a card numbered 1 for the first time. One integer N (1 ≤ N ≤ 106). One floating point number, the expected number of cards on the desk when a card with the number 1 is drawn for the first time. Your answer will be considered correct if the absolute and relative error compared with the judge's solution is less than 10 - 4. ## Input One integer N (1 ≤ N ≤ 106). ## Output One floating point number, the expected number of cards on the desk when a card with the number 1 is drawn for the first time. Your answer will be considered correct if the absolute and relative error compared with the judge's solution is less than 10 - 4. [samples]

**Definitions** Let $ G $ be the rotation group of the buckyball (truncated icosahedron), isomorphic to $ A_5 \times \mathbb{Z}_2 $, of order $ |G| = 60 $. Let $ X $ be the set of all functions $ f: \{1, \dots, 60\} \to \{0, 1, \dots, n\} $, where $ f(i) = 0 $ denotes no atom attached, and $ f(i) \in \{1, \dots, n\} $ denotes attachment of one of $ n $ atom types on carbon atom $ i $. Let $ G $ act on $ X $ by permuting the 60 carbon positions via spatial rotations. **Constraints** - $ n \in \mathbb{Z}_{\geq 0} $, representable as a signed 32-bit integer. - Two configurations are equivalent iff one is mapped to the other by some $ g \in G $. - Chirality is preserved: mirror images are distinct. **Objective** Compute the number of distinct orbits under the action of $ G $ on $ X $, i.e., $$ \frac{1}{|G|} \sum_{g \in G} n^{c(g)} $$ where $ c(g) $ is the number of cycles in the permutation induced by $ g $ on the 60 carbon atoms, and the result is taken modulo $ 1000000007 $.

API Response (JSON)

{
  "problem": {
    "name": "F. Drawing cards",
    "description": {
      "content": "You have a box with N cards numbered 1...N and you can also create new cards. You are going to draw cards from the box, every time you draw a card of a number that has already been drawn you put that",
      "description_type": "Markdown"
    },
    "platform": "Codeforces",
    "limit": {
      "time_limit": 1000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "CF10230F"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You have a box with N cards numbered 1...N and you can also create new cards.\n\nYou are going to draw cards from the box, every time you draw a card of a number that has already been drawn you put that...",
      "is_translate": false,
      "language": "English"
    },
    {
      "statement_type": "Markdown",
      "content": "**Definitions**  \nLet $ G $ be the rotation group of the buckyball (truncated icosahedron), isomorphic to $ A_5 \\times \\mathbb{Z}_2 $, of order $ |G| = 60 $.  \nLet $ X $ be the set of all functions $ ...",
      "is_translate": false,
      "language": "Formal"
    }
  ]
}

Full JSON Raw Segments