| Problem Statement |
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| There are n vertices forming a simple cycle. It is guaranteed that the graph satisfies the following property: |
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| - There exists a permutation p of length n such that for every i (1 ≤ i < n), the vertices p_i and p_{i+1} are adjacent in the graph, and also p_n and p_1 are adjacent. |
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| At the beginning of the interaction, a token is placed at a predetermined starting vertex s (1 ≤ s ≤ n). |
| The value of n is hidden from the contestant. The contestant may interact with the judge in order to determine n. |
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| Interaction Protocol |
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| The contestant may issue the following commands: |
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| 1. walk x |
| - Input: a nonnegative integer x (0 ≤ x ≤ 10^9). |
| - Effect: the token moves x steps forward along the cycle from its current position. |
| - Output: the label of the vertex reached after the move. |
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| 2. guess g |
| - Input: an integer g (1 ≤ g ≤ 10^9). |
| - Effect: ends the interaction. |
| - If g = n, the answer is considered correct. Otherwise, it is considered wrong. |
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| Constraints: |
| - 1 ≤ n ≤ 10^9 |
| - 1 ≤ s ≤ n |
| - The number of walk operations must not exceed 200000. |
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| Scoring |
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| Let q be the number of walk operations made before the guess. |
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| - If the guess is wrong or q > 200000, the score is 0. |
| - Otherwise, the score is f(q), where f is a continuous, monotonically decreasing function defined in log10-space by linear interpolation through the following anchor points: |
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| f(1) = 100 |
| f(10000) = 95 |
| f(20000) = 60 |
| f(50000) = 30 |
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| and further decreasing linearly to |
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| f(200000) = 0. |
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| Thus, fewer walk queries yield a higher score, with a perfect solution scoring close to 100. |
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